#6449010

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437798
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2257 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:

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

Polycyclic presentation:

2^9|a^2=f,d^2=g,f^2=i,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,f^b=fi,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, 6356206);