#411144

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

768 conjugacy classes of subgroups; only the 210 normal subgroups are shown.

normal subgroups only

This group has 768 conjugacy classes of subgroups — too many to draw. Only its 210 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)(17,18,23,19,21,25,22,27)(20,26,30,31,28,32,29,24);(2,6)(7,8)(9,10,12,14)(11,15,16,13)(17,19,22,25,21,27,23,18)(20,26,29,31,28,32,30,24);(1,3,5,4)(2,7,6,8)(9,11,12,16)(10,13,14,15)(17,20,23,30,21,28,22,29)(18,24,27,32,25,31,19,26);(1,4,5,3)(2,7,6,8)(17,20,22,29,21,28,23,30)(18,24,19,26,25,31,27,32)

Polycyclic presentation:

2^9|a^2=ef,b^2=gh,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, 318340);