#145141

order	512
nilpotency class	4
isoclinism	isoclinic class of #145131
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2033
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	#28240	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

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

normal subgroups only

This group has 348 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:

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

Polycyclic presentation:

2^9|c^2=dg,d^2=i,e^2=g,f^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eg,f^a=fi,c^b=cf,d^b=di,e^b=ei,f^b=fh,h^b=hi,e^c=egi,f^c=fhi,h^c=hi

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, 52337);