#72312491

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×B(4,𝔽₂)
commutator, Frattini, cosocle	C₂²×D₄	C₂⁵
LCS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×C₄×D₄ ◁ G
LECS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁

5905 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 32 points:

(1,2,6,9)(3,7,4,11)(5,13,14,8)(10,16,12,15)(17,18,22,21)(19,26,29,23)(20,25,31,24)(27,28,30,32);(1,3,6,4)(2,7,9,11)(5,12,14,10)(8,16,13,15)(17,19)(18,23)(20,30)(21,26)(22,29)(24,32)(25,28)(27,31);(1,4,6,3)(2,7,9,11)(5,12,14,10)(8,15,13,16)(17,20)(18,24)(19,27,29,30)(21,25)(22,31)(23,32,26,28);(1,5)(2,8)(3,10)(4,12)(6,14)(7,15)(9,13)(11,16)(17,21,22,18)(19,28,29,32)(20,24,31,25)(23,27,26,30);(5,14)(8,13)(10,12)(15,16)(19,29)(23,26)(27,30)(28,32)

Polycyclic presentation:

2^10|a^2=f,b^2=fj,c^2=fij,d^2=j,b^a=bj,c^a=cfi,c^b=ch,d^a=dfj,d^b=dfg,d^c=dj,e^b=ej,e^d=efj,g^c=gi,h^a=hj,h^c=hj,h^d=hij,h^g=hj,i^b=ij

Conjugacy classes & character table

88 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72312527, #72313107
maximal normal subgroup of	#72312527, #72313107

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