#6354144

order	512
nilpotency class	3
isoclinism	isoclinic class of #6354139
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#121
commutator, Frattini, cosocle	C₄²	C₂⁵
prime radical	C₂×#532	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1791 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6261340);