#143313

order	512
nilpotency class	4
isoclinism	isoclinic class of #143235
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	#33580	C₂
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1770 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 433 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:

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

Polycyclic presentation:

2^9|b^2=gh,d^2=gi,e^2=hi,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,f^a=fi,g^a=gi,h^a=hi,c^b=cf,d^b=dg,e^b=ehi,g^b=gi,h^b=hi,d^c=dh,e^c=eh,g^c=gi,h^c=hi

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, 50509);