#6449193

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437935
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#120
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3352 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)(11,22)(12,25)(13,19)(14,31)(15,21)(16,20)(17,24)(18,28)(23,36)(26,33)(27,35)(29,37)(30,34)(32,38)(39,40);(9,11,14,24,15,13,16,30)(10,17,20,22,21,34,31,19)(12,26,29,23,28,39,32,27)(18,35,37,40,25,36,38,33);(2,5)(6,8)(9,12)(10,18)(11,23)(13,27)(14,29)(15,28)(16,32)(17,33)(19,36)(20,37)(21,25)(22,35)(24,39)(26,30)(31,38)(34,40);(1,3,4,7)(2,6,5,8)(9,13,15,11)(10,19,21,22)(12,27,28,23)(14,30,16,24)(17,31,34,20)(18,36,25,35)(26,32,39,29)(33,38,40,37);(12,28)(18,25)(23,27)(26,39)(29,32)(33,40)(35,36)(37,38)

Polycyclic presentation:

2^9|b^2=f,d^2=g,f^2=i,h^2=i,b^a=bf,c^a=cg,f^a=fi,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, 6356389);