C₂×#15947 #609879

order	512
nilpotency class	2
isoclinism	isoclinic class of #15257
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹⁴

Aliases

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

27770 conjugacy classes of subgroups; only the 1322 normal subgroups are shown.

normal subgroups only

This group has 27770 conjugacy classes of subgroups — too many to draw. Only its 1322 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 517075);