#6351377

order	512
nilpotency class	3
isoclinism	isoclinic class of #18992
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

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

3124 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:

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

Polycyclic presentation:

2^9|e^2=h,f^2=h,g^2=hi,b^a=bf,c^a=cg,d^a=dh,f^a=fh,g^a=ghi,f^b=fh,g^b=gi,f^c=fi,g^c=ghi

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, 6258573);