#143246

order	512
nilpotency class	4
isoclinism	isoclinic class of #143217
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1999
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	#3147	C₂²
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#446 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 447 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:

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

Polycyclic presentation:

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

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, 50442);