#72323165

order	2048
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₁₆ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	#8144
commutator	C₂×C₈	C₂×C₄×C₁₆
Frattini, cosocle	C₂×#1397	C₂³
socle	C₂	#72286363
prime radical	C₂×#41	#346
LCS	G ▷ C₂×C₈ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂×#1397 ◁ G
LECS	G ▷ C₂×#1397 ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

664 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 128 points:

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

Polycyclic presentation:

2^11|a^2=d,b^2=hij,c^2=f,d^2=g,f^2=h,g^2=i,h^2=j,i^2=j,j^2=k,b^a=be,c^a=ch,d^c=dj,e^b=ek,f^a=fj,f^d=fk,g^c=gk,h^a=hk

Conjugacy classes & character table

260 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72323166, #72323173, #72323182
maximal normal subgroup of	#72323166, #72323173, #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.4);;
gap> SetPower(coll, 2, F.8*F.9*F.10);;
gap> SetPower(coll, 3, F.6);;
gap> SetPower(coll, 4, F.7);;
gap> SetPower(coll, 6, F.8);;
gap> SetPower(coll, 7, F.9);;
gap> SetPower(coll, 8, F.10);;
gap> SetPower(coll, 9, F.10);;
gap> SetPower(coll, 10, F.11);;
gap> SetConjugate(coll, 2, 1, F.2*F.5);;
gap> SetConjugate(coll, 3, 1, F.3*F.8);;
gap> SetConjugate(coll, 4, 3, F.4*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.11);;
gap> SetConjugate(coll, 6, 1, F.6*F.10);;
gap> SetConjugate(coll, 6, 4, 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);