#145187

order	512
nilpotency class	4
isoclinism	isoclinic class of #145131
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2033
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#452
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

297 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 297 conjugacy classes of subgroups — too many to draw. Only its 78 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) = 256

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=dg,d^2=i,e^2=g,f^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eg,f^a=fi,c^b=cf,d^b=di,e^b=ei,f^b=fh,h^b=hi,e^c=egi,f^c=fhi,h^c=hi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 52383);