C₉² #725

order	81
nilpotency class	1
isoclinism	isoclinic class of C₁
automorphism	Aut(G) = C₃×#13049364 (order 3888) · Inn(G) = C₁ · Out(G) = C₃×#13049364
homology	Abelianization (H₁) = G · Schur multiplier (H₂) = C₉

Aliases

C₉²

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, socle, prime radical, cosocle	C₃²	C₃²
LCS	G ▷ C₁
UCS	C₁ ◁ G
LECS	G ▷ C₃² ▷ C₁

23 conjugacy classes of subgroups.

minimal degree	μ(G) = 18
transitive degree	μ_t(G) = 81

A minimal faithful permutation representation — 2 generators on 18 points:

(1,2,3,4,5,6,7,8,9);(10,11,12,13,14,15,16,17,18)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(81, 2);