#145941

order	512
nilpotency class	4
isoclinism	isoclinic class of #145904
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂

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

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

normal subgroups only

This group has 223 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:

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

Polycyclic presentation:

2^9|a^2=de,b^2=dg,c^2=gi,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, 53137);