C₂×Q₈² #3477

order	128
nilpotency class	2
isoclinism	isoclinic class of D₄²
automorphism	Aut(G) order 1179648
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

C₂×Q₈²

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

556 conjugacy classes of subgroups.

minimal degree	μ(G) = 18
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 18 points:

(1,2);(3,4,5,6)(7,10,9,8);(3,7,5,9)(4,8,6,10);(11,12,13,14)(15,18,17,16);(11,15,13,17)(12,16,14,18)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(128, 2209);