#6575442

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573652
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 ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2046 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)(3,6)(4,5)(7,8)(9,10,15,13)(11,16,22,17)(12,23,25,20)(14,19,30,29)(18,27,28,26)(21,31,34,32)(24,36,37,35)(33,39,40,38);(2,5)(6,8)(9,11)(10,17)(12,18)(13,16)(14,19)(15,22)(20,27)(21,31)(23,26)(24,35)(25,28)(29,30)(32,34)(33,38)(36,37)(39,40);(9,12,15,25)(10,18,13,28)(11,20,22,23)(14,24,30,37)(16,27,17,26)(19,33,29,40)(21,35,34,36)(31,39,32,38);(1,3,4,7)(2,6,5,8)(9,13,15,10)(11,16,22,17)(12,23,25,20)(14,21,30,34)(18,26,28,27)(19,31,29,32)(24,33,37,40)(35,38,36,39);(9,14)(10,19)(11,21)(12,24)(13,29)(15,30)(16,31)(17,32)(18,33)(20,35)(22,34)(23,36)(25,37)(26,38)(27,39)(28,40)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6482638);