C₂×#10301 #233318

order	512
nilpotency class	3
isoclinism	isoclinic class of #10300
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

	subgroup	G/H
center, socle	C₂³	#392
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#3253	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1508 conjugacy classes of subgroups; only the 218 normal subgroups are shown.

normal subgroups only

This group has 1508 conjugacy classes of subgroups — too many to draw. Only its 218 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);(3,4,7,5)(6,8,10,9)(11,12)(13,16)(14,17)(15,18);(4,8)(5,9)(6,10)(11,12)(13,17)(14,16)(15,18);(3,5)(4,7)(6,8)(9,10)(12,16)(13,14)(17,18)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 140514);