C₂²×#2679 #7633434

order	512
nilpotency class	2
isoclinism	isoclinic class of #2668
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹⁴×C₄

Aliases

	subgroup	G/H
center, socle	C₂⁵	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂³ ▷ C₁

29842 conjugacy classes of subgroups; only the 4082 normal subgroups are shown.

normal subgroups only

This group has 29842 conjugacy classes of subgroups — too many to draw. Only its 4082 normal subgroups are shown.

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

minimal degree	μ(G) = 16
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 16 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7540630);