#115370

order	512
nilpotency class	3
isoclinism	isoclinic class of #115364
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1438
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

904 conjugacy classes of subgroups.

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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,d^2=h,e^2=h,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dh,f^a=fh,c^b=cf,d^b=dh,e^b=ehi,f^b=fh,d^c=di,e^c=eh,f^c=fh,g^c=gh

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, 22566);