C₁₁×#2397 #13065235

order	1408
nilpotency class	2
isoclinism	isoclinic class of #392
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₁₁ · Schur multiplier (H₂) = C₂⁶

Aliases

	subgroup	G/H
center	C₂²×C₄×C₁₁	C₂³
commutator, Frattini, cosocle	C₂³	C₂⁴×C₁₁
socle	C₂³×C₁₁	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₁₁ ◁ G

840 conjugacy classes of subgroups.

minimal degree	μ(G) = 27
transitive degree	μ_t(G) = 704

A minimal faithful permutation representation — 4 generators on 27 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1408, 1129);