C₂²×B(4,𝔽₂) #60392

order	256
nilpotency class	3
isoclinism	isoclinic class of B(4,𝔽₂)
automorphism	Aut(G) order 589824
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹¹

Aliases

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

3507 conjugacy classes of subgroups; only the 567 normal subgroups are shown.

normal subgroups only

This group has 3507 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);(9,12);(5,6)(7,9)(8,10)(11,12);(5,7)(6,9)(8,11)(10,12)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 53380);