#143271

order	512
nilpotency class	4
isoclinism	isoclinic class of #143211
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	#19101	C₂
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₈×D₄ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

419 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 419 conjugacy classes of subgroups — too many to draw. Only its 78 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)(11,19)(12,23)(13,28)(14,17)(15,20)(16,26)(18,25)(21,35)(22,37)(24,34)(27,38)(29,32)(30,36)(31,39)(33,40);(1,2,4,6)(3,8,7,5)(9,11)(10,15)(12,24)(13,29)(14,21)(16,32)(17,30)(18,34)(19,23)(20,26)(22,25)(27,39)(28,31)(33,37)(35,38)(36,40);(1,2)(3,8)(4,6)(5,7)(15,30)(19,35)(31,34)(32,37)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 50467);