#6454846

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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3001 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)(11,21)(12,17)(13,18)(14,28)(15,19)(16,24)(20,29)(22,35)(23,36)(25,37)(26,34)(27,33)(30,39)(31,38)(32,40);(9,11,14,24,15,25,29,40)(10,16,20,21,19,32,28,37)(12,23,26,39,13,22,27,38)(17,30,33,36,18,31,34,35);(2,5)(6,8)(9,12,15,13)(10,17,19,18)(11,22,25,23)(14,27,29,26)(16,30,32,31)(20,34,28,33)(21,35,37,36)(24,39,40,38);(9,13,15,12)(10,18,19,17)(11,23,25,22)(14,26,29,27)(16,31,32,30)(20,33,28,34)(21,36,37,35)(24,38,40,39);(1,3,4,7)(2,6,5,8)(10,19)(16,32)(17,18)(20,28)(21,37)(30,31)(33,34)(35,36)

Polycyclic presentation:

2^9|b^2=f,c^2=i,d^2=i,e^2=g,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, 6362042);