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 | #2431 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4356 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,4)(6,8)(25,26,31,33)(27,35,37,32)(28,38,40,29)(30,39,36,34);(3,7)(6,8)(9,10)(11,17)(12,20)(13,14)(15,19)(16,21)(18,23)(22,24)(25,27,31,37)(26,32,33,35)(28,39,40,34)(29,36,38,30);(1,2,5,4)(3,6,7,8)(10,14)(12,21)(15,17)(18,22)(26,33)(29,38)(30,36)(32,35);(9,11)(10,15)(12,22)(13,19)(14,17)(16,24)(18,21)(20,23)(25,28,31,40)(26,29,33,38)(27,34,37,39)(30,35,36,32);(1,3,5,7)(2,6,4,8)(9,12)(10,16)(11,18)(13,21)(14,20)(15,23)(17,24)(19,22)(25,29)(26,28)(27,36)(30,37)(31,38)(32,39)(33,40)(34,35);(1,4,5,2)(3,8,7,6)(9,10)(11,15)(12,16)(13,14)(17,19)(18,23)(20,21)(22,24)(25,30,31,36)(26,34,33,39)(27,29,37,38)(28,35,40,32)Polycyclic presentation:
2^9|a^2=g,b^2=g,c^2=gh,d^2=g,e^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,e^c=egi,f^c=fgi,e^d=egi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8948138);