#145925

order	512
nilpotency class	4
isoclinism	isoclinic class of #145904
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2067
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#464
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1450 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 247 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)(4,7)(6,8)(9,10,17,29,15,21,30,28,13,19,14,20,16,12,26,34)(11,22,32,33,31,27,38,40,23,18,24,36,25,35,39,37);(1,3,5,2,6,7,4,8)(9,11,13,23)(10,18,19,22)(12,27,21,35)(14,24,17,32)(15,31,16,25)(20,33,29,36)(26,39,30,38)(28,37,34,40);(2,3)(4,5)(7,8)(9,12,13,21)(10,15,19,16)(11,18,23,22)(14,29,17,20)(24,37,32,40)(25,35,31,27)(26,34,30,28)(33,39,36,38)

Polycyclic presentation:

2^9|a^2=de,b^2=dg,c^2=i,d^2=g,e^2=hi,f^2=gi,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eg,f^a=fg,c^b=cf,f^b=fi,d^c=dg,e^c=eh,f^c=fgi,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, 53121);