#143241

order	512
nilpotency class	4
isoclinism	isoclinic class of #143219
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	C₂×#2246	C₂
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2960 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 484 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)(12,15)(13,27)(14,19)(16,21)(17,24)(18,25)(22,31)(23,33)(26,30)(28,36)(29,40)(32,38)(34,39)(35,37);(3,7)(5,8)(10,15)(11,21)(12,24)(13,25)(16,31)(17,18)(19,30)(20,28)(22,38)(26,40)(27,29)(32,35)(33,34)(36,39);(1,2)(3,8)(4,6)(5,7)(9,11,13,23,14,21,25,37)(10,16,18,33,19,20,27,35)(12,22,24,39,26,36,40,38)(15,28,29,34,30,31,17,32)

Polycyclic presentation:

2^9|c^2=g,d^2=gi,e^2=h,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,g^b=gi,d^c=dh,e^c=ehi,f^c=fi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 50437);