#72321748

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 ◁ C₂×#7330 ◁ 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:

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

Polycyclic presentation:

2^11|a^2=e,c^2=f,d^2=gijk,e^2=g,f^2=h,g^2=i,i^2=j,j^2=k,b^a=bh,c^a=cijk,d^c=djk,e^b=ek,e^c=ej,f^a=fjk,f^d=fk,f^e=fk,g^c=gk,h^a=hk

Conjugacy classes & character table

260 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72322015, #72322689, #72322743
maximal normal subgroup of	#72322015, #72322689, #72322743

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.7*F.9*F.10*F.11);;
gap> SetPower(coll, 5, F.7);;
gap> SetPower(coll, 6, F.8);;
gap> SetPower(coll, 7, F.9);;
gap> SetPower(coll, 9, F.10);;
gap> SetPower(coll, 10, F.11);;
gap> SetConjugate(coll, 2, 1, F.2*F.8);;
gap> SetConjugate(coll, 3, 1, F.3*F.9*F.10*F.11);;
gap> SetConjugate(coll, 4, 3, F.4*F.10*F.11);;
gap> SetConjugate(coll, 5, 2, F.5*F.11);;
gap> SetConjugate(coll, 5, 3, F.5*F.10);;
gap> SetConjugate(coll, 6, 1, F.6*F.10*F.11);;
gap> SetConjugate(coll, 6, 4, F.6*F.11);;
gap> SetConjugate(coll, 6, 5, F.6*F.11);;
gap> SetConjugate(coll, 7, 3, F.7*F.11);;
gap> SetConjugate(coll, 8, 1, F.8*F.11);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);