C₂×C₁₁×#397 #13065228

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

Aliases

C₂×C₁₁×#397

	subgroup	G/H
center, socle	C₂⁴×C₁₁	C₂³
commutator, Frattini, cosocle	C₂³	C₂⁴×C₁₁
prime radical	C₂⁵×C₁₁	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₁₁ ◁ G

820 conjugacy classes of subgroups.

minimal degree	μ(G) = 29
transitive degree	μ_t(G) = 704

A minimal faithful permutation representation — 4 generators on 29 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1408, 1122);