#143735

order	512
nilpotency class	4
isoclinism	isoclinic class of #143705
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×#2206	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

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

normal subgroups only

This group has 445 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)(12,25)(13,16)(14,19)(15,22)(17,26)(18,27)(21,31)(23,30)(24,32)(28,35)(29,39)(33,38)(34,40)(36,37);(1,2,4,6)(3,8,7,5)(9,11,14,24)(10,15,19,31)(12,21,26,22)(13,23,28,39)(16,33,35,36)(17,32,25,20)(18,30,34,29)(27,37,40,38);(10,16)(11,21)(12,26)(13,27)(15,29)(17,18)(19,35)(20,36)(22,24)(23,39)(25,34)(28,40)(30,31)(32,33)

Polycyclic presentation:

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

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, 50931);