#143608

order	512
nilpotency class	4
isoclinism	isoclinic class of #143606
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	#19689	C₂
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1451 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

398 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 398 conjugacy classes of subgroups — too many to draw. Only its 68 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)(11,22)(12,27)(13,17)(14,19)(15,21)(16,24)(18,28)(20,29)(23,35)(25,38)(26,32)(30,36)(31,37)(33,40)(34,39);(1,2)(3,8)(4,6)(5,7)(9,11,15,26)(10,16,21,34)(12,23,30,40)(13,25,14,31)(17,35,19,33)(18,37,20,38)(22,36,32,27)(24,29,39,28);(1,2)(3,6)(4,8)(5,7)(10,17)(11,23)(12,28)(13,14)(16,32)(18,20)(19,21)(22,34)(24,25)(26,40)(29,30)(31,39)(33,37)(35,38)

Polycyclic presentation:

2^9|b^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=fhi,g^a=gi,h^a=hi,c^b=cf,d^b=dg,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

41 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 50804);