#9515324

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4341 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|b^2=gh,c^2=gh,d^2=h,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^c=egi,f^c=fgi,e^d=egi

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, 9422520);