Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2031 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | #124 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C24×C4 | C23 |
| socle | C23 | #394 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C2 ▷ C1 | |
1666 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 40 points:
(2,4)(5,6,7,8)(9,10)(11,21)(12,26)(13,20)(14,18)(15,19)(16,23)(17,28)(22,33)(24,31)(25,32)(27,35)(29,34)(30,39)(36,38)(37,40);(9,11,14,24,15,25,13,23)(10,16,20,32,19,31,18,21)(12,22,27,37,29,39,28,38)(17,33,34,36,35,30,26,40);(1,2,3,4)(9,12,13,28,15,29,14,27)(10,17,18,26,19,35,20,34)(11,22,23,38,25,39,24,37)(16,30,21,36,31,33,32,40)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=e,d^2=i,e^2=i,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=ei,f^a=fi,h^a=hi,c^b=cf,d^c=di104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 31089);