#72329893

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁵	C₂⁵
prime radical	C₂³×#120	C₂²
LCS	G ▷ C₂⁵ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₁

At least 6380 conjugacy classes of subgroups; only the 1129 normal subgroups are shown.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^10|b^2=fj,c^2=fj,d^2=fgh,b^a=bf,c^a=cg,c^b=cg,d^a=dfgi,d^c=dj,e^b=eg,e^c=eg,e^d=ehi

Conjugacy classes & character table

160 conjugacy classes — view the full character table.

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