C₂³×#392 #7627685

order	512
nilpotency class	2
isoclinism	isoclinic class of #392
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹⁷

Aliases

	subgroup	G/H
center, socle	C₂⁶	C₂³
commutator, Frattini, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁶ ◁ G
LECS	G ▷ C₂³ ▷ C₁

69930 conjugacy classes of subgroups; only the 7170 normal subgroups are shown.

normal subgroups only

This group has 69930 conjugacy classes of subgroups — too many to draw. Only its 7170 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) = 256

A minimal faithful permutation representation — 6 generators on 18 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7534881);