#6672955

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 ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1972 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|b^2=i,d^2=i,e^2=fi,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,h^b=hi,e^c=ei,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, 6580151);