#145144

order	512
nilpotency class	4
isoclinism	isoclinic class of #145143
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2034
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂²×D₄	SD₁₆
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

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

normal subgroups only

This group has 358 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) = 128

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

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

Polycyclic presentation:

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

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, 52340);