C₂×#2021 #23441

order	256
nilpotency class	3
isoclinism	isoclinic class of #2021
automorphism	Aut(G) order 65536
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁸

Aliases

	subgroup	G/H
center, socle	C₂³	#120
commutator, Frattini, cosocle	C₂⁴	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₂ ▷ C₁

1993 conjugacy classes of subgroups; only the 191 normal subgroups are shown.

normal subgroups only

This group has 1993 conjugacy classes of subgroups — too many to draw. Only its 191 normal subgroups are shown.

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

minimal degree	μ(G) = 14
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 4 generators on 14 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 16429);