#35390

order	256
nilpotency class	2
isoclinism	isoclinic class of #35346
automorphism	Aut(G) order 1572864
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

No aliases for this group.

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

2805 conjugacy classes of subgroups; only the 575 normal subgroups are shown.

normal subgroups only

This group has 2805 conjugacy classes of subgroups — too many to draw. Only its 575 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:

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

Polycyclic presentation:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 28378);