#72328811

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

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 6274 conjugacy classes of subgroups; only the 1003 normal subgroups are shown.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 36 points:

(6,9)(10,12)(13,14,17,18)(15,16,19,20)(21,22,25,26)(23,27,28,24)(29,30,33,35)(31,32,34,36);(1,2,3,4)(7,11)(10,12)(22,26)(24,27)(29,31)(30,32)(33,34)(35,36);(15,19)(16,20)(23,28)(24,27)(31,34)(32,36);(1,2,3,4)(5,6)(7,12)(8,9)(10,11)(13,15)(14,16)(17,19)(18,20)(21,23)(22,27)(24,26)(25,28)(29,31)(30,32)(33,34)(35,36);(1,2)(3,4)(5,7)(6,10)(8,11)(9,12)(13,16)(14,15)(17,20)(18,19)(21,24,25,27)(22,28,26,23)(29,32)(30,34)(31,35)(33,36)

Polycyclic presentation:

2^10|a^2=j,b^2=h,d^2=h,e^2=g,b^a=bg,c^b=cf,d^a=dhi,d^c=dj,e^a=efj,e^b=efgi,e^c=ej,e^d=efgi

Conjugacy classes & character table

142 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, 1, F.10);;
gap> SetPower(coll, 2, F.8);;
gap> SetPower(coll, 4, F.8);;
gap> SetPower(coll, 5, F.7);;
gap> SetConjugate(coll, 2, 1, F.2*F.7);;
gap> SetConjugate(coll, 3, 2, F.3*F.6);;
gap> SetConjugate(coll, 4, 1, F.4*F.8*F.9);;
gap> SetConjugate(coll, 4, 3, F.4*F.10);;
gap> SetConjugate(coll, 5, 1, F.5*F.6*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.6*F.7*F.9);;
gap> SetConjugate(coll, 5, 3, F.5*F.10);;
gap> SetConjugate(coll, 5, 4, F.5*F.6*F.7*F.9);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);