Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D42 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C42 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C8 | C24 |
| commutator | C22 | C23×C42 |
| Frattini, cosocle | C22×C4 | C25 |
| socle | C23 | C24×C4 |
| prime radical | C22×#41 | C2×C4 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2334 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(5,6)(7,10)(8,9)(11,12)(13,14,19,22,18,17,16,21)(15,23,24,28,25,20,26,27);(1,2,3,4)(6,9)(7,11)(13,15)(14,20)(16,26)(17,23)(18,25)(19,24)(21,28)(22,27);(5,7,8,11)(6,10,9,12)(13,16,18,19)(14,21,17,22)(15,24,25,26)(20,27,23,28);(6,9)(10,12)(15,25)(20,23)(24,26)(27,28);(13,17,19,21,18,14,16,22)(15,23,24,28,25,20,26,27)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=f,e^2=g,g^2=i,b^a=bf,d^a=dfi,c^b=cf,d^b=di200 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6277703);