#72267686

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁴×C₄³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁵	C₂⁵
prime radical	C₂⁶	C₂⁴
LCS	G ▷ C₂⁵ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₁

13996 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^10|a^2=f,b^2=fh,d^2=j,e^2=j,c^a=cfg,c^b=cgi,d^a=dfhij,d^b=dgh,e^b=eij,e^d=ej

Conjugacy classes & character table

142 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72267687
maximal normal subgroup of	#72267687

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