#410801

order	512
nilpotency class	3
isoclinism	isoclinic class of #395169
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

724 conjugacy classes of subgroups; only the 194 normal subgroups are shown.

normal subgroups only

This group has 724 conjugacy classes of subgroups — too many to draw. Only its 194 normal subgroups are shown.

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

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 32 points:

(1,2)(3,8)(4,7)(5,6)(9,10)(11,14)(12,13)(15,16)(17,18,24,19,22,27,23,30)(20,28,32,26,21,29,31,25);(2,6)(7,8)(10,13)(11,15)(17,19,23,27,22,30,24,18)(20,28,31,26,21,29,32,25);(1,3,5,4)(2,7,6,8)(9,11,12,15)(10,14,13,16)(17,20,24,32,22,21,23,31)(18,25,30,29,27,26,19,28);(1,4,5,3)(2,7,6,8)(17,21,23,32,22,20,24,31)(18,26,19,29,27,25,30,28)

Polycyclic presentation:

2^9|a^2=efg,b^2=h,c^2=fgi,d^2=f,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,e^b=ei,d^c=di

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 317997);