#6342401

order	512
nilpotency class	2
isoclinism	isoclinic class of #6342371
automorphism	Aut(G) order 1409286144
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄³

Aliases

No aliases for this group.

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

7878 conjugacy classes of subgroups; only the 1450 normal subgroups are shown.

normal subgroups only

This group has 7878 conjugacy classes of subgroups — too many to draw. Only its 1450 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) = 256

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

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

Polycyclic presentation:

2^9|b^2=f,c^2=g,d^2=h,b^a=bf,c^a=cg,d^a=dh,e^a=ei

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6249597);