Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2668 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C8 | C24 |
| commutator, socle | C23 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| prime radical | #52167 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2248 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2)(3,4)(5,6)(7,11)(8,10)(9,12)(13,14,19,23,18,17,22,28)(15,24,16,26,25,20,27,21);(5,7)(6,9)(8,12)(10,11)(13,15)(14,20)(16,19)(17,24)(18,25)(21,23)(22,27)(26,28);(2,4)(7,12)(9,11)(13,16)(14,21)(15,22)(17,26)(18,27)(19,25)(20,28)(23,24);(6,10)(9,11)(15,25)(16,27)(20,24)(21,26);(13,17,19,28,18,14,22,23)(15,24,16,26,25,20,27,21)Polycyclic presentation:
2^9|a^2=h,e^2=h,h^2=i,b^a=bf,c^a=cg,d^a=dfi,c^b=cf,d^b=di,d^c=di128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6284678);