C₂×#2679 #52098

order	256
nilpotency class	2
isoclinism	isoclinic class of #2668
automorphism	Aut(G) order 6291456
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹×C₄

Aliases

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

3311 conjugacy classes of subgroups; only the 583 normal subgroups are shown.

normal subgroups only

This group has 3311 conjugacy classes of subgroups — too many to draw. Only its 583 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) = 64

A minimal faithful permutation representation — 5 generators on 14 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 45086);