#72287111

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	#10580932	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ #3444 ◁ #60204 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

4517 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)(4,9)(5,19)(6,10)(7,11)(12,20)(13,28)(14,22)(15,21)(16,29)(17,23)(18,24)(25,32)(26,31)(27,30)(35,45)(36,47)(40,56)(41,60)(46,57)(48,63)(49,53)(59,62);(1,3,7,14)(2,8,11,22)(4,12,18,26)(5,13,6,15)(9,20,24,31)(10,21,19,28)(16,25,17,27)(23,30,29,32)(33,34,37,42)(35,41,45,60)(36,48,47,63)(38,52,50,61)(39,55,51,44)(40,57,56,46)(43,64,58,54)(49,62,53,59);(1,4,6,17,7,18,5,16)(2,9,10,23,11,24,19,29)(3,12,15,27,14,26,13,25)(8,20,21,30,22,31,28,32)(33,35,37,45)(34,40)(36,39,47,51)(38,53,50,49)(41,61)(42,56)(43,48)(44,62)(46,54,57,64)(52,60)(55,59)(58,63);(1,5,7,6)(2,10,11,19)(3,13,14,15)(4,16,18,17)(8,21,22,28)(9,23,24,29)(12,25,26,27)(20,30,31,32)(33,36,39,45,37,47,51,35)(34,41,44,63,42,60,55,48)(38,49,54,57,50,53,64,46)(40,58,59,61,56,43,62,52);(3,14)(8,22)(12,26)(13,15)(20,31)(21,28)(25,27)(30,32)(34,42)(35,45)(36,47)(43,58)(44,55)(46,57)(49,53)(52,61)

Polycyclic presentation:

2^10|b^2=hj,c^2=f,d^2=hi,f^2=h,g^2=ij,i^2=j,c^a=cj,c^b=cg,d^a=dhj,d^b=dij,d^c=di,e^b=ehj,e^c=ej,e^d=ej,f^b=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	#72287243, #72319631, #72319749
maximal normal subgroup of	#72287243, #72319631, #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.8*F.10);;
gap> SetPower(coll, 3, F.6);;
gap> SetPower(coll, 4, F.8*F.9);;
gap> SetPower(coll, 6, F.8);;
gap> SetPower(coll, 7, F.9*F.10);;
gap> SetPower(coll, 9, F.10);;
gap> SetConjugate(coll, 3, 1, F.3*F.10);;
gap> SetConjugate(coll, 3, 2, F.3*F.7);;
gap> SetConjugate(coll, 4, 1, F.4*F.8*F.10);;
gap> SetConjugate(coll, 4, 2, F.4*F.9*F.10);;
gap> SetConjugate(coll, 4, 3, F.4*F.9);;
gap> SetConjugate(coll, 5, 2, F.5*F.8*F.10);;
gap> SetConjugate(coll, 5, 3, F.5*F.10);;
gap> SetConjugate(coll, 5, 4, F.5*F.10);;
gap> SetConjugate(coll, 6, 2, 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);