#72237067

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#3334833
commutator, Frattini, cosocle	C₂³×C₄	C₂⁵
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

5499 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^10|b^2=f,c^2=fhj,d^2=i,h^2=j,b^a=bf,c^a=cfg,c^b=ci,d^a=dgj,d^b=dh,d^c=dgj,e^b=egi,e^c=egh,e^d=eij,f^c=fj,f^d=fj,g^b=gj,g^c=gj,h^c=hj,h^e=hj,i^b=ij

Conjugacy classes & character table

76 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72237665, #72251940, #72251951
maximal normal subgroup of	#72237665, #72251940, #72251951

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