C₂²×#392 #34531

order	256
nilpotency class	2
isoclinism	isoclinic class of #392
automorphism	Aut(G) order 75497472
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹²

Aliases

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

6823 conjugacy classes of subgroups; only the 983 normal subgroups are shown.

normal subgroups only

This group has 6823 conjugacy classes of subgroups — too many to draw. Only its 983 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 — 5 generators on 16 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 27519);