#6491425

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470058
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#120
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2972 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6398621);