#72317499

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#8546
commutator	C₂²×#41	C₂²×C₄
Frattini, cosocle	C₂³×#41	C₂³
socle	C₂	#111516
prime radical	#6431743	C₂
LCS	G ▷ C₂²×#41 ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂³×#41 ◁ G
LECS	G ▷ C₂³×#41 ▷ C₂³ ▷ C₂ ▷ C₁

4393 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 64 points:

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

Polycyclic presentation:

2^10|a^2=e,b^2=h,c^2=fij,e^2=j,f^2=j,b^a=bd,c^a=cg,c^b=cf,d^a=dh,e^b=eh,e^c=ei,g^a=gi,g^b=gj,g^d=gj,h^c=hj,i^b=ij

Conjugacy classes & character table

76 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72317506, #72317535, #72317714
maximal normal subgroup of	#72317506, #72317535, #72317714

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