#6583825

order	512
nilpotency class	3
isoclinism	isoclinic class of #6574075
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

1699 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:

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

Polycyclic presentation:

2^9|a^2=fi,c^2=g,d^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=di,e^b=ehi,h^b=hi,d^c=dh,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, 6491021);