Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2010 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | #127 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C24×C4 | C23 |
| socle | C23 | #392 |
| prime radical | C25 | #41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C2 ▷ C1 | |
1642 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2,3,4)(9,10)(11,21)(12,25)(13,19)(14,18)(15,20)(16,23)(17,26)(22,32)(24,34)(27,35)(28,31)(29,33)(30,39)(36,38)(37,40);(5,6,7,8)(9,11)(10,16)(12,22)(13,28)(14,23)(15,24)(17,32)(18,34)(19,21)(20,31)(25,40)(26,38)(27,39)(29,37)(30,33)(35,36);(9,12,14,26,15,27,13,29)(10,17,19,25,20,33,18,35)(11,22,23,38,24,39,28,37)(16,30,21,36,31,32,34,40)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=e,d^2=i,e^2=i,b^a=bd,c^a=ce,d^a=di,e^a=ei,f^a=fi,c^b=cf,d^b=di,d^c=di104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 31096);