#72265407

order	6561
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₃³ · Schur multiplier (H₂) = C₃⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₃²	#10624562
commutator, Frattini, cosocle	C₃⁵	C₃³
LCS	G ▷ C₃⁵ ▷ C₃² ▷ C₁
UCS	C₁ ◁ C₃² ◁ C₃⁵ ◁ G
LECS	G ▷ C₃⁵ ▷ C₃² ▷ C₁

6106 conjugacy classes of subgroups; only the 108 normal subgroups are shown.

normal subgroups only

This group has 6106 conjugacy classes of subgroups — too many to draw. Only its 108 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 54
transitive degree	μ_t(G) = 243

A minimal faithful permutation representation — 3 generators on 54 points:

(1,2,5)(3,6,12)(4,7,14)(8,16,22)(9,17,21)(10,15,23)(11,18,13)(19,24,27)(20,25,26)(28,29,32)(30,33,40)(31,35,41)(34,45,39)(36,42,48)(37,44,49)(38,43,50)(46,52,53)(47,51,54);(6,15,17)(12,21,23)(16,25,24)(22,27,26)(28,30,36)(29,33,42)(31,38,46)(32,40,48)(34,44,51)(35,43,52)(37,47,39)(41,50,53)(45,49,54);(1,3,8)(2,6,16)(4,10,19)(5,12,22)(7,15,24)(9,20,11)(13,21,26)(14,23,27)(17,25,18)(33,43,44)(40,49,50)(42,51,52)(48,53,54)

Polycyclic presentation:

3^8|b^a=bd,c^a=cf,c^b=ce,d^c=dg,e^a=eg^2h,f^b=fh

Conjugacy classes & character table

161 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72265524
maximal normal subgroup of	#72265524
maximal quotient of	#72265419

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");;
gap> coll := SingleCollector(F, [3, 3, 3, 3, 3, 3, 3, 3]);;
gap> SetConjugate(coll, 2, 1, F.2*F.4);;
gap> SetConjugate(coll, 3, 1, F.3*F.6);;
gap> SetConjugate(coll, 3, 2, F.3*F.5);;
gap> SetConjugate(coll, 4, 3, F.4*F.7);;
gap> SetConjugate(coll, 5, 1, F.5*F.7^2*F.8);;
gap> SetConjugate(coll, 6, 2, F.6*F.8);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);