#72318517

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	#10580907	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

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

Polycyclic presentation:

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

Conjugacy classes & character table

142 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72318591, #72318672, #72319190
maximal normal subgroup of	#72318591, #72318672, #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, 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*F.10);;
gap> SetConjugate(coll, 3, 2, F.3*F.7);;
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);