Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #53390 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C25 |
| commutator, Frattini, socle, cosocle | C23 | C26 |
| prime radical | #54928 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4426 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,6)(3,4,7,8)(9,10,15,17)(11,19,22,16)(12,13,18,14)(20,21,24,23)(25,26)(27,30)(28,33)(29,34)(31,35)(32,37)(36,38)(39,40);(1,3,5,7)(2,4,6,8)(9,11)(10,16)(12,20)(13,23)(14,21)(15,22)(17,19)(18,24)(25,27,30,26)(28,33,35,31)(29,32,37,34)(36,38,40,39);(1,4)(2,3)(5,8)(6,7)(9,12)(10,13)(11,20)(14,17)(15,18)(16,23)(19,21)(22,24)(25,28)(26,31)(27,33)(29,36)(30,35)(32,38)(34,39)(37,40);(3,7)(4,8)(10,17)(12,18)(16,19)(20,24)(26,27)(28,35)(32,34)(36,40);(2,6)(3,7)(9,13,15,14)(10,18,17,12)(11,21,22,23)(16,20,19,24)(25,29,30,37)(26,32,27,34)(28,36,35,40)(31,38,33,39);(9,14,15,13)(10,12,17,18)(11,21,22,23)(16,20,19,24)Polycyclic presentation:
2^9|a^2=gh,b^2=h,e^2=g,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,d^c=dgi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8012730);