C₂×#10300 #233317

order	512
nilpotency class	3
isoclinism	isoclinic class of #10300
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

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

1622 conjugacy classes of subgroups; only the 218 normal subgroups are shown.

normal subgroups only

This group has 1622 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)(5,7)(6,9)(8,10)(12,16)(13,14)(17,18);(4,7)(5,10)(8,9)(11,12)(13,18)(14,16)(15,17);(3,4)(5,8)(6,9)(7,10)(11,12)(13,16)(14,18)(15,17)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 140513);