C₂⁴×SD₃₂ #10585863

order	512
nilpotency class	4
isoclinism	isoclinic class of D₁₆
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹⁴

Aliases

C₂⁴×SD₃₂

	subgroup	G/H
center, socle	C₂⁵	D₈
commutator, Frattini, cosocle	C₈	C₂⁶
prime radical	C₂⁴×D₈	C₂
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ C₂⁴×C₄ ◁ C₂⁴×C₈ ◁ G
LECS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

9106 conjugacy classes of subgroups.

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 24 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);(10,16)(11,23)(12,14)(13,21)(15,19)(18,24)(20,22)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10493059);