#72307269

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#25502
commutator, Frattini, cosocle	C₂³×C₈	C₂⁴
socle	C₂	#403908
LCS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂²×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

4486 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:

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

Polycyclic presentation:

2^10|b^2=fj,c^2=e,d^2=f,e^2=i,g^2=ij,i^2=j,b^a=bf,c^a=ch,c^b=cg,d^c=degj,e^b=ei,e^d=ej,f^c=fj,g^b=gij,g^c=gj,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.5);;
gap> SetPower(coll, 4, F.6);;
gap> SetPower(coll, 5, F.9);;
gap> SetPower(coll, 7, F.9*F.10);;
gap> SetPower(coll, 9, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.6);;
gap> SetConjugate(coll, 3, 1, F.3*F.8);;
gap> SetConjugate(coll, 3, 2, F.3*F.7);;
gap> SetConjugate(coll, 4, 3, F.4*F.5*F.7*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.9);;
gap> SetConjugate(coll, 5, 4, F.5*F.10);;
gap> SetConjugate(coll, 6, 3, F.6*F.10);;
gap> SetConjugate(coll, 7, 2, F.7*F.9*F.10);;
gap> SetConjugate(coll, 7, 3, F.7*F.10);;
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);