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 | #53490 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4376 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,4,6)(3,7,8,5)(9,10,14,17)(11,18,20,15)(12,16,22,13)(19,21,24,23)(25,26,30,33)(27,29,35,31)(28,36,38,32)(34,37,40,39);(1,3)(2,5)(4,8)(6,7)(9,11)(10,15)(12,19)(13,21)(14,20)(16,23)(17,18)(22,24)(25,27,30,35)(26,31,33,29)(28,34,38,40)(32,37,36,39);(3,8)(5,7)(9,12)(10,16)(11,19)(13,17)(14,22)(15,23)(18,21)(20,24)(25,28,30,38)(26,32,33,36)(27,34,35,40)(29,39,31,37);(2,6)(5,7)(10,17)(12,22)(15,18)(19,24)(25,29,30,31)(26,27,33,35)(28,37,38,39)(32,40,36,34);(11,20)(15,18)(19,24)(21,23)(26,33)(27,35)(32,36)(34,40);(9,13,14,16)(10,12,17,22)(11,21,20,23)(15,19,18,24)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=h,d^2=h,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, 8012829);