#72312191

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 ◁ C₂×C₄² ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁

5848 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:

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

Polycyclic presentation:

2^10|b^2=f,c^2=fij,d^2=fhj,e^2=fhi,b^a=bf,c^a=ch,c^b=cfhj,d^a=dij,d^b=dfi,d^c=dhij,e^b=eg,e^c=eg,e^d=efhj,f^c=fj,f^d=fj,f^e=fi,g^a=gij,g^b=gi,g^c=gi,g^d=gi,g^f=gj,h^b=hj,h^d=hj,h^e=hi,h^g=hj,i^b=ij,i^c=ij

Conjugacy classes & character table

88 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72312232, #72312260, #72312586, …
maximal normal subgroup of	#72312232, #72312260, #72312586, …

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