C₃×#6879 #10624835

order	729
nilpotency class	4
isoclinism	isoclinic class of #6878
automorphism	?
homology	Abelianization (H₁) = C₃³ · Schur multiplier (H₂) = C₃³×C₉

Aliases

	subgroup	G/H
center, socle	C₃²	#732
commutator, Frattini, cosocle	C₃×C₉	C₃³
LCS	G ▷ C₃×C₉ ▷ C₃² ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₃² ◁ C₃³ ◁ C₃²×C₉ ◁ G
LECS	G ▷ C₃×C₉ ▷ C₃² ▷ C₃ ▷ C₁

130 conjugacy classes of subgroups.

minimal degree	μ(G) = 30
transitive degree	μ_t(G) = 81

A minimal faithful permutation representation — 3 generators on 30 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(729, 395);