C₂³×#393 #7627686

order	512
nilpotency class	2
isoclinism	isoclinic class of #392
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹⁶

Aliases

	subgroup	G/H
center, socle	C₂⁶	C₂³
commutator, Frattini, cosocle	C₂³	C₂⁶
prime radical	C₂⁷	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁶ ◁ G
LECS	G ▷ C₂³ ▷ C₁

33466 conjugacy classes of subgroups; only the 6658 normal subgroups are shown.

normal subgroups only

This group has 33466 conjugacy classes of subgroups — too many to draw. Only its 6658 normal subgroups are shown.

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

minimal degree	μ(G) = 22
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 22 points:

(1,2);(3,4);(5,6);(12,14)(15,16,18,20)(17,21,22,19);(8,10)(12,14)(15,17,18,22)(16,19,20,21);(7,8)(9,10)(11,12)(13,14)(15,16)(17,21)(18,20)(19,22)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7534882);