#145185

order	512
nilpotency class	4
isoclinism	isoclinic class of #145129
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

294 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 294 conjugacy classes of subgroups — too many to draw. Only its 76 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)(11,21,25,26)(12,13,18,14)(15,19,33,32)(17,28,36,23)(22,31,35,34)(24,39,40,38)(27,29,37,30);(2,5)(6,7)(9,11,16,25)(10,17,20,36)(12,26,18,21)(13,28,14,23)(15,31,33,34)(19,38,32,39)(22,37,35,27)(24,30,40,29);(2,6)(4,8)(5,7)(9,12,14,10,16,18,13,20)(11,22,28,39,25,35,23,38)(15,32,30,27,33,19,29,37)(17,34,26,40,36,31,21,24)

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,c^b=cf,d^b=di,f^b=fh,h^b=hi,e^c=egi,f^c=fhi,h^c=hi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 52381);