#6445198

order	512
nilpotency class	3
isoclinism	isoclinic class of #6439035
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

1937 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)(9,10,15,21)(11,23,26,17)(12,27,31,33)(13,19,35,28)(14,20,16,22)(18,30,39,32)(24,37,40,34)(25,38,36,29);(2,5)(6,8)(9,11)(10,17)(12,28)(13,33)(14,25)(15,26)(16,36)(18,34)(19,31)(20,29)(21,23)(22,38)(24,32)(27,35)(30,40)(37,39);(9,12,16,30,15,31,14,32)(10,18,20,33,21,39,22,27)(11,24,36,28,26,40,25,19)(13,29,37,23,35,38,34,17);(9,13)(10,19)(11,18)(12,29)(14,34)(15,35)(16,37)(17,32)(20,24)(21,28)(22,40)(23,30)(25,27)(26,39)(31,38)(33,36);(1,3,4,7)(2,6,5,8)(9,14,15,16)(10,20,21,22)(11,25,26,36)(12,30,31,32)(13,34,35,37)(17,29,23,38)(18,27,39,33)(19,24,28,40)

Polycyclic presentation:

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

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, 6352394);