#145940

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

Polycyclic presentation:

2^9|a^2=de,b^2=dg,c^2=g,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, 53136);