#6467249

order	512
nilpotency class	3
isoclinism	isoclinic class of #6442590
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

1923 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|b^2=i,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,f^b=fi,h^b=hi,d^c=di,e^c=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, 6374445);