#6454868

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437969
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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2964 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)(3,6)(4,5)(7,8)(9,10,15,19)(11,20,18,14)(12,23,25,16)(13,17,28,32)(21,29,33,26)(22,27,34,30)(24,36,37,31)(35,38,40,39);(9,11,14,10,15,18,20,19)(12,21,26,23,25,33,29,16)(13,22,27,32,28,34,30,17)(24,35,38,31,37,40,39,36);(2,5)(6,8)(9,12,15,25)(10,16,19,23)(11,21,18,33)(13,24,28,37)(14,29,20,26)(17,31,32,36)(22,35,34,40)(27,39,30,38);(9,13,15,28)(10,17,19,32)(11,22,18,34)(12,24,25,37)(14,30,20,27)(16,31,23,36)(21,35,33,40)(26,39,29,38);(1,3,4,7)(2,6,5,8)(12,25)(16,23)(21,33)(24,37)(26,29)(31,36)(35,40)(38,39)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=i,d^2=i,e^2=gi,f^2=i,b^a=bf,c^a=cg,f^a=fi,h^a=hi,c^b=ch,d^b=dh,h^b=hi,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, 6362064);