#7554

order	256
nilpotency class	2
isoclinism	isoclinic class of #120
automorphism	Aut(G) order 786432
homology	Abelianization (H₁) = C₄³ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

	subgroup	G/H
center, Frattini, socle, prime radical, cosocle	C₂⁵	C₂³
commutator	C₂²	C₄³
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₁

1579 conjugacy classes of subgroups; only the 607 normal subgroups are shown.

normal subgroups only

This group has 1579 conjugacy classes of subgroups — too many to draw. Only its 607 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 3 generators on 20 points:

(6,8)(9,10,12,11)(18,19);(13,14,16,15)(17,18,20,19);(1,2,3,4)(5,6,7,8)(9,11,12,10)(13,15,16,14)(17,19,20,18)

Polycyclic presentation:

2^8|a^2=f,b^2=g,c^2=h,b^a=bd,c^a=ce

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 542);