C₂⁴×C₄⋊Q₈ #10585896

order	512
nilpotency class	2
isoclinism	isoclinic class of #120
automorphism	?
homology	Abelianization (H₁) = C₂⁷ · Schur multiplier (H₂) = C₂²⁰

Aliases

C₂⁴×C₄⋊Q₈

	subgroup	G/H
center, socle, prime radical	C₂⁶	C₂³
commutator, Frattini, cosocle	C₂²	C₂⁷
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁶ ◁ G
LECS	G ▷ C₂² ▷ C₁

62338 conjugacy classes of subgroups.

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 7 generators on 20 points:

(1,2);(3,4);(5,6);(7,8);(10,12)(13,14,17,20)(15,19,16,18);(13,15,17,16)(14,18,20,19);(9,10,11,12)(13,16,17,15)(14,19,20,18)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10493092);