C₂×#13461 #499213

order	512
nilpotency class	4
isoclinism	isoclinic class of #13343
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

	subgroup	G/H
center, socle	C₂³	B(4,𝔽₂)
commutator, Frattini, cosocle	C₂²×D₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴ ◁ C₂³×D₄ ◁ G
LECS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁

1468 conjugacy classes of subgroups; only the 202 normal subgroups are shown.

normal subgroups only

This group has 1468 conjugacy classes of subgroups — too many to draw. Only its 202 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 — 4 generators on 18 points:

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

maximal subgroup of	C₂×#499281, #72281660, #72281714
maximal normal subgroup of	C₂×#499281, #72281660, #72281714

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 406409);