C₂×#2029 #23465

order	256
nilpotency class	3
isoclinism	isoclinic class of #2021
automorphism	Aut(G) order 196608
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

	subgroup	G/H
center, socle	C₂³	#120
commutator, Frattini, cosocle	C₂⁴	C₂⁴
prime radical	C₂⁶	C₂²
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₂ ▷ C₁

1409 conjugacy classes of subgroups; only the 175 normal subgroups are shown.

normal subgroups only

This group has 1409 conjugacy classes of subgroups — too many to draw. Only its 175 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);(3,4,6,8)(5,9,10,7)(11,12)(13,15)(14,16)(17,18);(3,5,6,10)(4,7,8,9)(11,13)(12,15)(14,17)(16,18);(15,18)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 16453);