#145014

order	512
nilpotency class	4
isoclinism	isoclinic class of #145011
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2041
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

388 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 388 conjugacy classes of subgroups — too many to draw. Only its 80 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)(3,5)(4,7)(6,8)(9,10,16,22)(11,14,25,32)(12,23,29,39)(13,21,15,19)(17,20,30,24)(18,34,36,28)(26,31,35,40)(27,37,33,38);(9,11,16,25)(10,17,22,30)(12,26,29,35)(13,20,15,24)(14,19,32,21)(18,27,36,33)(23,34,39,28)(31,38,40,37);(2,5)(3,8)(6,7)(9,12,13,27,16,29,15,33)(10,18,19,35,22,36,21,26)(11,23,24,38,25,39,20,37)(14,31,30,28,32,40,17,34)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=d,d^2=i,e^2=gi,f^2=hi,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eg,f^a=fh,g^a=gi,h^a=hi,c^b=cf,d^b=di,e^b=eh,f^b=fh,g^b=gi,h^b=hi,e^c=egi,f^c=fhi,g^c=gi,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, 52210);