#6448770

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437452
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#120
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#60425	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3175 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)(11,18)(12,17)(13,19)(14,27)(15,20)(16,22)(21,33)(23,30)(24,34)(25,36)(26,35)(28,31)(29,39)(32,37)(38,40);(9,11)(10,16)(12,21)(13,22)(14,28)(15,23)(17,32)(18,27)(19,30)(20,31)(24,37)(25,39)(26,38)(29,35)(33,36)(34,40);(2,5)(6,8)(9,12,15,26)(10,17,20,35)(11,21,23,38)(13,25,14,24)(16,29,31,32)(18,33,30,40)(19,36,27,34)(22,39,28,37);(1,3,4,7)(2,6,5,8)(9,11)(10,18)(12,21)(13,22)(14,28)(15,23)(16,19)(17,33)(20,30)(24,37)(25,39)(26,38)(27,31)(29,36)(32,34)(35,40);(9,13,15,14)(10,19,20,27)(11,22,23,28)(12,24,26,25)(16,30,31,18)(17,34,35,36)(21,37,38,39)(29,33,32,40)

Polycyclic presentation:

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

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, 6355966);