#6445416

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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2932 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=g,d^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, 6352612);