#143625

order	512
nilpotency class	4
isoclinism	isoclinic class of #143618
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2011
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#493
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₈×M₁₆ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 381 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=i,d^2=gi,e^2=gh,f^2=hi,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=egh,f^a=fh,g^a=gi,h^a=hi,c^b=cf,d^b=dg,e^b=ei,f^b=fh,g^b=gi,h^b=hi,d^c=dh,e^c=egh,f^c=fh,g^c=gi,h^c=hi

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 50821);