#72319269

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#442
commutator	C₂×C₈	C₂⁴×C₄
Frattini, cosocle	C₄×C₈	C₂⁵
prime radical	#7623523	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ #17323 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

2341 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 64 points:

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

Polycyclic presentation:

2^10|c^2=hj,d^2=f,f^2=h,g^2=ij,i^2=j,b^a=bh,c^a=chj,c^b=cg,d^a=dh,d^b=dh,d^c=di,e^b=eh,e^d=ehj,f^c=fj,g^b=gi,g^c=gi,g^d=gj,i^b=ij,i^c=ij

Conjugacy classes & character table

106 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72319316, #72319621, #72319631
maximal normal subgroup of	#72319316, #72319621, #72319631

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