C₂²×#550 #62700

order	256
nilpotency class	2
isoclinism	isoclinic class of D₄²
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹⁴

Aliases

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

6459 conjugacy classes of subgroups; only the 3223 normal subgroups are shown.

normal subgroups only

This group has 6459 conjugacy classes of subgroups — too many to draw. Only its 3223 normal subgroups are shown.

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

minimal degree	μ(G) = 16
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 16 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 55688);