#168533

order	512
nilpotency class	3
isoclinism	isoclinic class of #168497
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#2682	C₂
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

1282 conjugacy classes of subgroups; only the 160 normal subgroups are shown.

normal subgroups only

This group has 1282 conjugacy classes of subgroups — too many to draw. Only its 160 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) = 128

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=gh,e^2=g,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=eg,f^a=fi,e^b=eg,f^b=fh,e^c=eh,f^c=fi

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 75729);