#72265249

order	6561
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₃⁴ · Schur multiplier (H₂) = C₃⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₃²	C₃×C₃²⋊H(3,𝔽₃)
commutator, Frattini, cosocle	C₃⁴	C₃⁴
LCS	G ▷ C₃⁴ ▷ C₃² ▷ C₁
UCS	C₁ ◁ C₃² ◁ C₃⁵ ◁ G
LECS	G ▷ C₃⁴ ▷ C₃² ▷ C₁

12768 conjugacy classes of subgroups; only the 345 normal subgroups are shown.

normal subgroups only

This group has 12768 conjugacy classes of subgroups — too many to draw. Only its 345 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) = 81

A minimal faithful permutation representation — 4 generators on 54 points:

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

Polycyclic presentation:

3^8|b^a=be,c^b=cf,d^a=dg^2,d^c=dh^2,e^b=eg,f^b=fh

Conjugacy classes & character table

225 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72265524
maximal normal subgroup of	#72265524
maximal quotient of	#72265253

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