Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C26 |
| prime radical | #55909 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4200 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,5)(3,4)(9,10,15,12)(11,14,17,18)(13,16,22,20)(19,21,23,24)(26,31)(28,34)(32,38)(36,40);(1,2,6,5)(3,8,4,7)(9,11)(10,14)(12,18)(13,19)(15,17)(16,21)(20,24)(22,23)(27,35)(32,38)(36,40)(37,39);(1,3,6,4)(2,7,5,8)(11,17)(14,18)(19,23)(21,24)(25,26)(27,32)(28,33)(29,34)(30,31)(35,38)(36,39)(37,40);(1,4,6,3)(2,8,5,7)(9,12,15,10)(11,18,17,14)(13,16,22,20)(19,21,23,24)(25,27)(26,32)(28,36)(29,37)(30,35)(31,38)(33,39)(34,40);(1,5,6,2)(3,8,4,7)(9,13)(10,16)(11,19)(12,20)(14,21)(15,22)(17,23)(18,24)(25,28,30,34)(26,33,31,29)(27,36,35,40)(32,39,38,37);(9,14,15,18)(10,17,12,11)(13,21,22,24)(16,23,20,19)(25,29,30,33)(26,34,31,28)(27,37,35,39)(32,40,38,36)Polycyclic presentation:
2^9|a^2=gi,b^2=g,c^2=g,d^2=i,e^2=h,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8241694);