#15142

order	256
nilpotency class	2
isoclinism	isoclinic class of #14715
automorphism	Aut(G) order 3145728
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶×C₄²

Aliases

No aliases for this group.

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

2191 conjugacy classes of subgroups; only the 295 normal subgroups are shown.

normal subgroups only

This group has 2191 conjugacy classes of subgroups — too many to draw. Only its 295 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 — 4 generators on 16 points:

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

Polycyclic presentation:

2^8|b^2=e,c^2=f,d^2=h,b^a=be,c^a=cf,d^a=dh,c^b=cg

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 8130);