#9062097

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

4452 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8969293);