#6348498

order	512
nilpotency class	3
isoclinism	isoclinic class of #6348493
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	C₄²	C₂⁵
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

3036 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)(6,8)(9,10)(11,20)(12,17)(13,18)(14,28)(15,19)(16,23)(21,34)(22,30)(24,33)(25,31)(26,32)(27,38)(29,37)(35,40)(36,39);(3,7)(5,8)(9,11,15,24)(10,16,19,30)(12,21,25,35)(13,23,14,22)(17,29,31,39)(18,33,28,20)(26,37,27,36)(32,40,38,34);(1,2)(3,8)(4,6)(5,7)(9,12)(10,17)(11,21)(13,26)(14,27)(15,25)(16,29)(18,32)(19,31)(20,34)(22,36)(23,37)(24,35)(28,38)(30,39)(33,40);(12,25)(17,31)(21,35)(26,27)(29,39)(32,38)(34,40)(36,37);(9,13,15,14)(10,18,19,28)(11,22,24,23)(12,26,25,27)(16,20,30,33)(17,32,31,38)(21,36,35,37)(29,34,39,40)

Polycyclic presentation:

2^9|b^2=h,e^2=h,f^2=h,g^2=i,b^a=bf,c^a=cg,f^a=fh,g^a=gi,e^b=eh,f^b=fh,g^b=gi,d^c=dh,f^c=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, 6255694);