#6467863

order	512
nilpotency class	3
isoclinism	isoclinic class of #6443048
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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1772 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)(11,22)(12,18)(13,19)(14,30)(15,20)(16,21)(17,25)(23,33)(24,39)(26,34)(27,32)(28,37)(29,38)(31,35)(36,40);(9,11,14,25,16,26,15,31)(10,17,20,22,21,35,30,34)(12,27,29,40,28,24,13,23)(18,36,19,32,37,33,38,39);(2,5)(6,8)(9,12,15,13,16,28,14,29)(10,18,20,19,21,37,30,38)(11,23,25,27,26,40,31,24)(17,32,34,36,35,39,22,33);(9,13,15,28,16,29,14,12)(10,19,20,37,21,38,30,18)(11,24,25,23,26,27,31,40)(17,33,34,32,35,36,22,39);(1,3,4,7)(2,6,5,8)(12,28)(13,29)(18,37)(19,38)(23,40)(24,27)(32,39)(33,36)

Polycyclic presentation:

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

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, 6375059);