C₂×D₈ #132

order	32
nilpotency class	3
isoclinism	isoclinic class of D₈
automorphism	Aut(G) = #23882 (order 256) · Inn(G) = D₄ · Out(G) = #120
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

C₂×D₈

	subgroup	G/H
center, socle	C₂²	D₄
commutator, Frattini, cosocle	C₄	C₂³
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ G
LECS	G ▷ C₄ ▷ C₂ ▷ C₁

38 conjugacy classes of subgroups.

minimal degree	μ(G) = 10
transitive degree	μ_t(G) = 16

A minimal faithful permutation representation — 3 generators on 10 points:

(1,2);(3,4)(5,7)(6,9)(8,10);(4,7)(5,10)(8,9)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(32, 39);