#145081

order	512
nilpotency class	4
isoclinism	isoclinic class of #145027
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2034
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#449
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

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

normal subgroups only

This group has 275 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) = 128

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

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

Polycyclic presentation:

2^9|a^2=hi,b^2=g,c^2=d,e^2=gi,f^2=h,g^2=i,b^a=bd,c^a=ce,e^a=eg,f^a=fh,g^a=gi,c^b=cf,d^b=di,e^b=eh,f^b=fhi,e^c=eg,f^c=fhi,g^c=gi

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, 52277);