#72328409

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁵×C₄³

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 6792 conjugacy classes of subgroups; only the 921 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:

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

Polycyclic presentation:

2^10|a^2=j,b^2=h,c^2=j,d^2=i,b^a=bg,c^a=cj,c^b=cf,d^a=dhi,d^b=dhi,d^c=dij,e^a=efg,e^b=efi,e^c=egi,e^d=egi

Conjugacy classes & character table

124 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, 3, F.10);;
gap> SetPower(coll, 4, F.9);;
gap> SetConjugate(coll, 2, 1, F.2*F.7);;
gap> SetConjugate(coll, 3, 1, F.3*F.10);;
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*F.9);;
gap> SetConjugate(coll, 4, 3, F.4*F.9*F.10);;
gap> SetConjugate(coll, 5, 1, F.5*F.6*F.7);;
gap> SetConjugate(coll, 5, 2, F.5*F.6*F.9);;
gap> SetConjugate(coll, 5, 3, F.5*F.7*F.9);;
gap> SetConjugate(coll, 5, 4, F.5*F.7*F.9);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);