#14693

order	256
nilpotency class	2
isoclinism	isoclinic class of #2846
automorphism	Aut(G) order 1572864
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

	subgroup	G/H
center, Frattini, socle, cosocle	C₂⁴	C₂⁴
commutator	C₂³	C₂³×C₄
prime radical	C₂⁶	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3247 conjugacy classes of subgroups; only the 415 normal subgroups are shown.

normal subgroups only

This group has 3247 conjugacy classes of subgroups — too many to draw. Only its 415 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) = 64

A minimal faithful permutation representation — 4 generators on 16 points:

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

Polycyclic presentation:

2^8|a^2=h,b^2=h,b^a=be,c^a=cf,d^a=dg

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 7681);