#72307261

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#25502
commutator, Frattini, cosocle	C₂³×C₈	C₂⁴
socle	C₂	#404277
prime radical	#10575331	C₂
LCS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂²×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

4147 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 64 points:

(9,30)(17,44)(26,51)(33,50)(35,57)(45,49)(53,63)(58,61);(1,2,7,11)(3,8,14,27)(4,17)(5,22,18,46)(6,24,25,12)(9,31)(10,34,32,23)(13,35)(15,41,39,62)(16,43,40,29)(19,44)(20,30)(21,49)(26,52)(28,55,54,42)(33,48)(36,57)(37,51)(38,61)(45,47)(50,56)(53,60)(58,59)(63,64);(1,3)(2,8)(4,13,19,36)(5,15)(6,16)(7,14)(9,26,30,51)(10,28)(11,27)(12,29)(17,35,44,57)(18,39)(20,37,31,52)(21,38,47,59)(22,41)(23,42)(24,43)(25,40)(32,54)(33,53,50,63)(34,55)(45,58,49,61)(46,62)(48,60,56,64);(1,4,18,31,6,21,34,56,7,19,5,20,25,47,23,48)(2,9,32,44,12,33,46,49,11,30,10,17,24,50,22,45)(3,13,15,37,16,38,42,60,14,36,39,52,40,59,55,64)(8,26,28,35,29,53,41,58,27,51,54,57,43,63,62,61)

Polycyclic presentation:

2^10|b^2=fj,c^2=f,d^2=efgj,g^2=i,i^2=j,b^a=bf,d^a=dh,d^b=dg,d^c=de,e^c=ej,e^d=ej,f^d=fj,g^b=gi,h^b=hj,i^b=ij

Conjugacy classes & character table

100 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72307305, #72307810
maximal normal subgroup of	#72307305, #72307810

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