#449475

order	512
nilpotency class	3
isoclinism	isoclinic class of #417493
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂¹⁰

Aliases

No aliases for this group.

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂⁵	C₂⁴
LCS	G ▷ C₂⁵ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁶ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

11222 conjugacy classes of subgroups; only the 266 normal subgroups are shown.

normal subgroups only

This group has 11222 conjugacy classes of subgroups — too many to draw. Only its 266 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 — 4 generators on 16 points:

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

Polycyclic presentation:

2^9|a^2=h,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,d^b=de,h^b=hi

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 356671);