C₂²×C₃×#494 #10679237

order	768
nilpotency class	3
isoclinism	isoclinic class of #465
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₃ · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

C₂²×C₃×#494

	subgroup	G/H
center, socle, prime radical	C₂⁴×C₃	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₃
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₃ ◁ C₂²×C₃×C₄² ◁ G

3262 conjugacy classes of subgroups.

minimal degree	μ(G) = 27
transitive degree	μ_t(G) = 768

A minimal faithful permutation representation — 5 generators on 27 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 53618);