C₂×#2848 #36612

order	256
nilpotency class	2
isoclinism	isoclinic class of #2846
automorphism	Aut(G) order 6291456
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹⁰

Aliases

	subgroup	G/H
center, socle	C₂⁴	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁵
prime radical	C₂⁶	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3751 conjugacy classes of subgroups; only the 751 normal subgroups are shown.

normal subgroups only

This group has 3751 conjugacy classes of subgroups — too many to draw. Only its 751 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) = 64

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

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 29600);