C₂×#2613 #34553

order	256
nilpotency class	2
isoclinism	isoclinic class of #2613
automorphism	Aut(G) order 4194304
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

3973 conjugacy classes of subgroups; only the 607 normal subgroups are shown.

normal subgroups only

This group has 3973 conjugacy classes of subgroups — too many to draw. Only its 607 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);(8,10)(11,12)(13,16)(14,15)(17,18);(4,6)(8,10)(12,15)(16,17);(3,4)(5,6)(7,8)(9,10)(11,12)(13,17)(14,15)(16,18);(11,13)(12,16)(14,18)(15,17)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 27541);