C₃×#15278 #10633885

order	768
nilpotency class	2
isoclinism	isoclinic class of #15257
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₃ · Schur multiplier (H₂) = C₂⁸

Aliases

	subgroup	G/H
center, socle	C₂⁴×C₃	C₂⁴
commutator, Frattini, cosocle	C₂⁴	C₂⁴×C₃
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₃ ◁ G

3134 conjugacy classes of subgroups.

minimal degree	μ(G) = 19
transitive degree	μ_t(G) = 384

A minimal faithful permutation representation — 4 generators on 19 points:

(1,2,3)(5,7)(8,9,10,11)(12,13)(14,15);(13,14)(17,19);(8,9)(10,11)(12,13,15,14)(16,17)(18,19);(4,5)(6,7)(8,9)(10,11)(12,14,15,13)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 8266);