#6670358

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667760
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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1918 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:

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

Polycyclic presentation:

2^9|a^2=i,c^2=i,d^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,g^a=gi,c^b=ch,e^b=efh,h^b=hi,g^c=gi,h^c=hi,h^e=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, 6577554);