C₁₁×#137 #67030

order	352
nilpotency class	3
isoclinism	isoclinic class of AGL(1,ℤ/8)
automorphism	Aut(G) = C₂×C₅×D₄² (order 640) · Inn(G) = C₂×D₄ · Out(G) = C₂³×C₅
homology	Abelianization (H₁) = C₂³×C₁₁ · Schur multiplier (H₂) = C₂²

Aliases

	subgroup	G/H
center, socle	C₂×C₁₁	C₂×D₄
commutator, Frattini, cosocle	C₄	C₂³×C₁₁
prime radical	C₁₁×#41	C₂
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₁ ◁ C₂×C₄×C₁₁ ◁ G

60 conjugacy classes of subgroups.

minimal degree	μ(G) = 27
transitive degree	μ_t(G) = 176

A minimal faithful permutation representation — 3 generators on 27 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(352, 172);