C₂×#2388 #34535

order	256
nilpotency class	2
isoclinism	isoclinic class of #392
automorphism	Aut(G) order 3145728
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹⁰

Aliases

	subgroup	G/H
center	C₂³×C₄	C₂³
commutator, Frattini, cosocle	C₂³	C₂⁵
socle	C₂⁴	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3119 conjugacy classes of subgroups; only the 639 normal subgroups are shown.

normal subgroups only

This group has 3119 conjugacy classes of subgroups — too many to draw. Only its 639 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) = 128

A minimal faithful permutation representation — 5 generators on 18 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 27523);