#6445797

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437963
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	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2888 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=f,c^2=gi,e^2=i,f^2=i,b^a=bf,c^a=cg,e^a=ei,f^a=fi,h^a=hi,c^b=ch,d^b=dh,h^b=hi,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, 6352993);