#6445669

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437798
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1997 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)(3,7)(4,5)(6,8)(9,10,16,22,15,11,14,19)(12,20,27,23,25,17,26,33)(13,18,31,34,30,21,29,28)(24,35,39,36,37,32,38,40);(2,5)(6,7)(9,11,15,10)(12,23,25,33)(13,28,30,34)(14,19,16,22)(17,27,20,26)(18,31,21,29)(24,35,37,32)(36,39,40,38);(1,3,4,8)(2,6,5,7)(9,12,15,25)(10,17,11,20)(13,24,30,37)(14,26,16,27)(18,32,21,35)(19,23,22,33)(28,36,34,40)(29,38,31,39);(9,13)(10,18)(11,21)(12,24)(14,29)(15,30)(16,31)(17,32)(19,28)(20,35)(22,34)(23,36)(25,37)(26,38)(27,39)(33,40);(12,25)(17,20)(23,33)(24,37)(26,27)(32,35)(36,40)(38,39)

Polycyclic presentation:

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

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, 6352865);