#145164

order	512
nilpotency class	4
isoclinism	isoclinic class of #145147
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2034
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

355 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 355 conjugacy classes of subgroups — too many to draw. Only its 76 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)(6,7)(10,16)(11,21)(12,14)(13,27)(17,32)(18,35)(19,20)(22,40)(23,28)(24,38)(25,39)(26,29)(30,33)(34,36);(1,2)(3,5)(4,6)(7,8)(9,10)(11,22)(12,19)(13,18)(14,16)(15,20)(17,23)(21,36)(24,34)(25,30)(26,37)(27,31)(28,40)(29,35)(32,39)(33,38);(2,6)(4,8)(5,7)(9,11,15,25)(10,17)(12,24,14,23)(13,28,29,38)(16,30)(18,36)(19,22)(20,34)(21,26,39,27)(31,33)(32,35)(37,40)

Polycyclic presentation:

2^9|c^2=dgh,d^2=i,e^2=gi,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eg,f^a=fi,g^a=gi,c^b=cf,d^b=di,f^b=fh,d^c=di,e^c=eg,f^c=fhi,g^c=gi

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, 52360);