C₂²×#453 #60378

order	256
nilpotency class	3
isoclinism	isoclinic class of #453
automorphism	Aut(G) order 393216
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹⁰

Aliases

	subgroup	G/H
center, socle	C₂³	#120
commutator, Frattini, cosocle	C₂×C₄	C₂⁵
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁

2835 conjugacy classes of subgroups; only the 567 normal subgroups are shown.

normal subgroups only

This group has 2835 conjugacy classes of subgroups — too many to draw. Only its 567 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

minimal degree	μ(G) = 12
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 5 generators on 12 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 53366);