#72287623

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	#10580912	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄×#41 ◁ C₂×C₈×#41 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

4673 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:

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

Polycyclic presentation:

2^10|a^2=f,c^2=j,f^2=h,g^2=i,i^2=j,c^b=ch,d^c=dh,e^a=eg,e^b=eh,e^c=ej,e^d=ej,f^e=fj,g^a=gi,g^e=gij,i^a=ij,i^e=ij

Conjugacy classes & character table

142 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72287657, #72287726, #72319190
maximal normal subgroup of	#72287657, #72287726, #72319190

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