#145898

order	512
nilpotency class	4
isoclinism	isoclinic class of #145862
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2058
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#450
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₈×C₄⋊C₄ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

257 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 257 conjugacy classes of subgroups — too many to draw. Only its 82 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)(3,4)(5,7)(6,8)(9,10,12,16)(11,21)(13,27,26,37)(14,20,15,19)(17,33)(18,29,32,30)(22,36)(23,31)(24,38)(25,39)(28,34)(35,40);(1,3,6,7)(2,4,8,5)(10,16)(11,22,25,24)(17,34,31,35)(18,32)(19,20)(21,36,39,38)(23,40,33,28)(27,37);(2,3)(4,5)(7,8)(9,11,14,24,12,25,15,22)(10,17,19,34,16,31,20,35)(13,28,29,33,26,40,30,23)(18,36,37,39,32,38,27,21)

Polycyclic presentation:

2^9|a^2=d,b^2=dhi,c^2=dgh,e^2=g,f^2=hi,g^2=i,b^a=bd,c^a=ce,e^a=eg,f^a=fhi,g^a=gi,c^b=cf,e^b=eh,f^b=fh,d^c=di,e^c=eg,f^c=fhi,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, 53094);