#6467858

order	512
nilpotency class	3
isoclinism	isoclinic class of #6443050
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 ◁ C₂×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1748 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)(12,17)(13,18)(14,20)(15,21)(16,24)(19,29)(23,28)(25,33)(26,34)(27,40)(30,35)(31,37)(32,38)(36,39);(9,11,14,24,15,17,29,25)(10,16,19,22,21,33,20,12)(13,27,31,23,30,38,39,34)(18,28,36,40,35,26,37,32);(2,5)(6,8)(9,12,14,16,15,22,29,33)(10,17,20,24,21,11,19,25)(13,28,31,40,30,26,39,32)(18,23,37,27,35,34,36,38);(9,13,29,39,15,30,14,31)(10,18,19,36,21,35,20,37)(11,23,24,38,17,34,25,27)(12,26,33,40,22,28,16,32);(1,3,4,7)(2,6,5,8)(13,30)(18,35)(23,34)(26,28)(27,38)(31,39)(32,40)(36,37)

Polycyclic presentation:

2^9|b^2=f,c^2=h,d^2=hi,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^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, 6375054);