D₈×C₁₉ #64535

order	304
nilpotency class	3
isoclinism	isoclinic class of D₈
automorphism	Aut(G) = C₂×C₉×AGL(1,ℤ/8) (order 576) · Inn(G) = D₄ · Out(G) = C₂³×C₉
homology	Abelianization (H₁) = C₂²×C₁₉ · Schur multiplier (H₂) = C₂

Aliases

D₈×C₁₉

	subgroup	G/H
center, socle	C₂×C₁₉	D₄
commutator, Frattini, cosocle	C₄	C₂²×C₁₉
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₉ ◁ C₄×C₁₉ ◁ G

22 conjugacy classes of subgroups.

minimal degree	μ(G) = 27
transitive degree	μ_t(G) = 152

A minimal faithful permutation representation — 2 generators on 27 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(304, 24);