#143720

order	512
nilpotency class	4
isoclinism	isoclinic class of #143719
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1999
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	#19189	C₂
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2931 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 446 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) = 256

A minimal faithful permutation representation — 3 generators on 40 points:

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

Polycyclic presentation:

2^9|b^2=i,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=ei,f^b=fi,g^b=gi,d^c=di,e^c=eh,f^c=fi

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, 50916);