Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #453 |
| automorphism | ? |
| homology | Abelianization (H1) = C43 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | #120 |
| commutator | C2×C4 | C43 |
| Frattini, cosocle | C24×C4 | C23 |
| socle | C23 | #385 |
| prime radical | C25 | C2×Q8 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C2 ▷ C1 | |
1730 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)(9,10,13,19,15,20,14,18)(11,21,25,30,24,31,23,16)(12,17,27,32,28,34,26,33)(22,36,37,29,38,35,39,40);(9,11,14,23,15,24,13,25)(10,16,18,31,20,30,19,21)(12,22,26,39,28,38,27,37)(17,29,33,36,34,40,32,35);(5,6,7,8)(9,12)(10,17)(11,22)(13,27)(14,26)(15,28)(16,29)(18,33)(19,32)(20,34)(21,35)(23,39)(24,38)(25,37)(30,40)(31,36)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=h,d^2=i,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=di,d^b=di,e^b=ei,d^c=di128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 30687);