#6583363

order	512
nilpotency class	3
isoclinism	isoclinic class of #6574077
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#135	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1636 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=f,c^2=gi,g^2=i,h^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=di,e^b=eh,h^b=hi,d^c=dhi,h^c=hi,e^d=eh,h^d=hi,h^e=hi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6490559);