#72328534

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	#156684	C₂
LCS	G ▷ C₂⁵ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₁

At least 6841 conjugacy classes of subgroups; only the 963 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)(13,14,16,18)(15,19,20,17)(22,25)(26,27)(29,30,33,35)(31,34,36,32);(1,2,3,4)(7,11)(10,12)(21,22)(23,26)(24,25)(27,28)(29,31,33,36)(30,34,35,32);(15,20)(17,19)(31,36)(32,34);(2,4)(5,6)(7,12)(8,9)(10,11)(13,14)(15,19)(16,18)(17,20)(21,22)(23,27)(24,25)(26,28)(29,30)(31,32)(33,35)(34,36);(1,2)(3,4)(5,7)(6,10)(8,11)(9,12)(13,15)(14,17)(16,20)(18,19)(21,23,24,28)(22,26,25,27)(29,32)(30,31)(33,34)(35,36)

Polycyclic presentation:

2^10|a^2=j,b^2=h,e^2=g,b^a=bg,c^b=cf,d^a=dhi,d^b=dh,e^a=efj,e^b=efgij,e^c=ej,e^d=ei

Conjugacy classes & character table

130 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, 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, 2, F.4*F.8);;
gap> SetConjugate(coll, 5, 1, F.5*F.6*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.6*F.7*F.9*F.10);;
gap> SetConjugate(coll, 5, 3, F.5*F.10);;
gap> SetConjugate(coll, 5, 4, F.5*F.9);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);