#9057791

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

4588 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=ghi,d^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,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, 8964987);