C₂×#2010 #23413

order	256
nilpotency class	3
isoclinism	isoclinic class of #2010
automorphism	Aut(G) order 16384
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

	subgroup	G/H
center	C₂×C₄	#127
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴
socle	C₂²	#392
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1095 conjugacy classes of subgroups; only the 183 normal subgroups are shown.

normal subgroups only

This group has 1095 conjugacy classes of subgroups — too many to draw. Only its 183 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) = 32

A minimal faithful permutation representation — 4 generators on 18 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 16401);