#72307619

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, socle	C₂	#346522
commutator, Frattini, cosocle	C₂×#131	C₂⁴
LCS	G ▷ C₂×#131 ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×#131 ◁ G
LECS	G ▷ C₂×#131 ▷ C₂×C₄ ▷ C₂ ▷ C₁

3567 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:

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

Polycyclic presentation:

2^10|b^2=h,c^2=e,d^2=h,e^2=ij,f^2=ij,i^2=j,b^a=bj,c^a=cg,c^b=cf,d^a=dh,d^c=defgh,e^a=eh,e^b=ehij,f^b=fij,f^c=fh,f^d=fh,f^e=fj,g^c=gh,g^d=gh,g^e=gj,h^c=hj,i^b=ij

Conjugacy classes & character table

73 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72307633, #72307752, #72307865
maximal normal subgroup of	#72307633, #72307752, #72307865

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