#146113

order	512
nilpotency class	4
isoclinism	isoclinic class of #146112
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2075
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂²×D₄	D₈
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₈×D₄ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

227 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 227 conjugacy classes of subgroups — too many to draw. Only its 66 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 — 3 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=e,c^2=di,d^2=g,e^2=hi,h^2=i,b^a=bd,c^a=ce,f^a=fg,c^b=cf,d^b=dg,e^b=eg,e^c=eh,f^c=fg,h^c=hi

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, 53309);