#72323167

order	2048
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₈ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	C₂×#1450
commutator	C₂×C₈	C₂²×C₄×C₈
Frattini, cosocle	#1397	C₂⁴
socle	C₂	C₂×#149513
prime radical	C₂×#41	#346
LCS	G ▷ C₂×C₈ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ #149496 ◁ G
LECS	G ▷ #1397 ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

820 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 128
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 128 points:

(1,2,6,16,8,18,36,71,10,20,34,15,53,75,37,76,11,21,5,39,42,82,35,72,9,56,44,77,55,17,38,81,12,22,45,80,48,74,115,127,57,85,43,78,7,49,50,70,51,67,46,68,41,118,54,14,58,19,84,69,52,83,4,33)(3,23,26,79,28,88,106,122,30,90,107,73,110,119,105,126,31,91,25,98,100,128,101,86,29,64,102,124,111,87,24,92,32,13,47,61,104,63,95,123,112,65,93,60,27,108,96,121,103,66,40,116,99,125,94,59,113,89,117,120,109,62,97,114);(1,3)(2,13)(4,24)(5,40)(6,47)(7,27)(8,28)(9,29)(10,30)(11,31)(12,32)(14,59)(15,73)(16,79)(17,62)(18,63)(19,64)(20,65)(21,66)(22,23)(25,46)(26,45)(33,114)(34,93)(35,94)(36,95)(37,96)(38,97)(39,98)(41,99)(42,100)(43,107)(44,117)(48,104)(49,119)(50,105)(51,103)(52,109)(53,110)(54,101)(55,111)(56,89)(57,112)(58,113)(60,78)(61,80)(67,91)(68,116)(69,120)(70,121)(71,122)(72,86)(74,88)(75,108)(76,126)(77,124)(81,92)(82,125)(83,87)(84,102)(85,90)(106,115)(118,128)(123,127);(1,4,7,34,9,35,8,45,51,50,55,84,10,36,41,5,12,38,53,43,58,54,48,6,11,37,52,44,57,115,42,46)(2,14,17,68,19,70,82,78,67,127,75,80,85,81,18,77,22,72,83,39,56,76,118,15,21,71,49,16,20,33,74,69)(3,24,27,93,29,94,28,26,103,105,111,102,30,95,99,40,32,97,110,107,113,101,104,47,31,96,109,117,112,106,100,25)(13,59,62,116,64,121,125,60,91,123,108,61,90,92,63,124,23,86,87,98,89,126,128,73,66,122,119,79,65,114,88,120);(1,5,41,115,57,84,55,37,11,45,8,54,58,34,7,38,12,46,42,36,10,44,52,50,51,6,48,35,9,43,53,4)(2,15,74,76,20,39,49,72,21,77,118,81,56,80,83,127,22,78,18,70,85,68,75,14,67,69,82,33,19,16,17,71)(3,25,99,95,112,117,111,105,31,47,28,94,113,107,27,24,32,40,100,106,30,102,109,96,103,26,104,101,29,93,110,97)(13,60,88,121,65,116,119,59,66,120,128,114,89,79,87,122,23,73,63,126,90,98,108,86,91,124,125,92,64,61,62,123)

Polycyclic presentation:

2^11|a^2=e,c^2=f,d^2=fi,e^2=g,f^2=h,g^2=i,h^2=jk,i^2=j,j^2=k,b^a=bhik,c^a=chj,d^a=dhj,d^b=dk,d^c=dk,e^b=ek,e^c=ej,e^d=ej,f^a=fj,f^e=fk,g^c=gk,g^d=gk,h^a=hk

Conjugacy classes & character table

260 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72323172, #72323177, #72323182
maximal normal subgroup of	#72323172, #72323177, #72323182

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e", "f", "g", "h", "i", "j", "k");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 1, F.5);;
gap> SetPower(coll, 3, F.6);;
gap> SetPower(coll, 4, F.6*F.9);;
gap> SetPower(coll, 5, F.7);;
gap> SetPower(coll, 6, F.8);;
gap> SetPower(coll, 7, F.9);;
gap> SetPower(coll, 8, F.10*F.11);;
gap> SetPower(coll, 9, F.10);;
gap> SetPower(coll, 10, F.11);;
gap> SetConjugate(coll, 2, 1, F.2*F.8*F.9*F.11);;
gap> SetConjugate(coll, 3, 1, F.3*F.8*F.10);;
gap> SetConjugate(coll, 4, 1, F.4*F.8*F.10);;
gap> SetConjugate(coll, 4, 2, F.4*F.11);;
gap> SetConjugate(coll, 4, 3, F.4*F.11);;
gap> SetConjugate(coll, 5, 2, F.5*F.11);;
gap> SetConjugate(coll, 5, 3, F.5*F.10);;
gap> SetConjugate(coll, 5, 4, F.5*F.10);;
gap> SetConjugate(coll, 6, 1, F.6*F.10);;
gap> SetConjugate(coll, 6, 5, F.6*F.11);;
gap> SetConjugate(coll, 7, 3, F.7*F.11);;
gap> SetConjugate(coll, 7, 4, F.7*F.11);;
gap> SetConjugate(coll, 8, 1, F.8*F.11);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);