#6574957

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
prime radical	#61055	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1980 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)(11,18)(12,13)(14,17)(15,31)(16,20)(19,28)(21,37)(22,26)(23,24)(25,33)(27,29)(30,32)(34,35)(36,38)(39,40);(2,5)(6,8)(9,11,17,25)(10,18,14,33)(12,26,30,39)(13,22,32,40)(15,24,16,34)(19,36,27,37)(20,35,31,23)(21,28,38,29);(1,3,4,7)(2,6,5,8)(9,12)(10,19)(11,21)(13,31)(14,27)(15,29)(16,28)(17,30)(18,22)(20,32)(23,37)(24,39)(25,38)(26,34)(33,40)(35,36);(9,13,17,32)(10,12,14,30)(11,22,25,40)(15,27,16,19)(18,26,33,39)(20,28,31,29)(21,35,38,23)(24,37,34,36);(9,14)(10,17)(11,23)(12,27)(13,29)(15,31)(16,20)(18,24)(19,30)(21,37)(22,39)(25,35)(26,40)(28,32)(33,34)(36,38)

Polycyclic presentation:

2^9|b^2=i,c^2=f,d^2=i,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, 6482153);