#143736

order	512
nilpotency class	4
isoclinism	isoclinic class of #143705
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1999
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂×#493	C₂²
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2960 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

429 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 429 conjugacy classes of subgroups — too many to draw. Only its 82 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) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 40 points:

(2,5)(3,7)(6,8)(11,20)(12,25)(13,14)(16,30)(17,32)(18,29)(21,38)(22,36)(23,26)(24,37)(27,28)(31,39)(33,35)(34,40);(1,2,4,6)(3,8,7,5)(9,10)(11,21)(12,17)(13,29)(14,18)(15,19)(16,23)(20,33)(22,35)(24,31)(25,32)(26,39)(27,40)(28,34)(30,37)(36,38);(9,11,15,22)(10,16,19,31)(12,26,28,37)(13,24,14,23)(17,33,34,38)(18,35,29,21)(20,27,36,25)(30,40,39,32)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 50932);