C₂×#28443 #6465524

order	512
nilpotency class	3
isoclinism	isoclinic class of #24378
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

	subgroup	G/H
center, socle	C₂⁴	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×#493	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂×C₄³ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

4050 conjugacy classes of subgroups.

minimal degree	μ(G) = 18
transitive degree	μ_t(G) = 256

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

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

maximal subgroup of	C₂×D₄×#463, C₂×#6467353, C₂×#6578236
maximal normal subgroup of	C₂×D₄×#463, C₂×#6467353, C₂×#6578236

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6372720);