#72318862

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂³×#102
commutator	C₂×C₈	C₂⁴×C₄
Frattini, cosocle	C₄×C₈	C₂⁵
prime radical	#10580937	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ #3444 ◁ #60204 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

4520 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)(3,8)(4,16)(5,18)(6,9)(7,11)(10,21)(12,28)(13,26)(14,22)(15,24)(17,25)(19,23)(20,27)(29,31)(30,32)(33,34)(35,39)(36,42)(37,43)(38,40)(41,49)(44,62)(45,57)(46,56)(47,58)(48,55)(50,61)(51,54)(52,60)(53,59)(63,64);(1,3,6,14,7,15,4,12)(2,8,9,22,11,24,16,28)(5,19,20,29,21,13,17,30)(10,26,25,32,18,23,27,31)(33,35,36,45,38,46,49,48)(34,39,42,57,40,56,41,55)(37,51,50,47,53,44,52,63)(43,54,61,58,59,62,60,64);(1,4,7,6)(2,9,11,16)(3,12,15,14)(5,20,21,17)(8,22,24,28)(10,27,18,25)(13,30,19,29)(23,32,26,31)(34,40)(35,44)(36,49)(37,52)(39,54)(43,61)(45,47)(46,51)(48,63)(50,53)(55,58)(56,62)(57,64)(59,60);(1,5)(2,10)(3,13)(4,17)(6,20)(7,21)(8,23)(9,25)(11,18)(12,29)(14,30)(15,19)(16,27)(22,31)(24,26)(28,32)(33,36,38,49)(34,41,40,42)(35,45,46,48)(37,50,53,52)(39,55,56,57)(43,60,59,61)(44,63,51,47)(54,64,62,58);(5,21)(10,18)(13,19)(17,20)(23,26)(25,27)(29,30)(31,32)(35,46)(39,56)(44,51)(45,48)(47,63)(54,62)(55,57)(58,64)

Polycyclic presentation:

2^10|b^2=f,c^2=hj,d^2=j,f^2=h,g^2=i,i^2=j,c^a=ch,c^b=cg,d^a=dh,d^b=dhj,d^c=dh,e^b=ej,e^d=ehj,f^c=fj,g^b=gi,g^c=gij,i^b=ij,i^c=ij

Conjugacy classes & character table

130 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72318898, #72318999, #72319749
maximal normal subgroup of	#72318898, #72318999, #72319749

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