#36614

order	256
nilpotency class	2
isoclinism	isoclinic class of #2846
automorphism	Aut(G) order 1572864
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹⁰

Aliases

No aliases for this group.

	subgroup	G/H
center	C₂²×C₄	C₂⁴
commutator, Frattini, socle, cosocle	C₂³	C₂⁵
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3769 conjugacy classes of subgroups; only the 631 normal subgroups are shown.

normal subgroups only

This group has 3769 conjugacy classes of subgroups — too many to draw. Only its 631 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) = 32

A minimal faithful permutation representation — 5 generators on 16 points:

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

Polycyclic presentation:

2^8|e^2=f,b^a=bf,c^a=cg,d^a=dh

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 29602);