Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(5,𝔽2) |
| automorphism | ? |
| homology | Abelianization (H1) = C28 · Schur multiplier (H2) = C227 |
| direct product | C23×#585 |
| subgroup | G/H | |
|---|---|---|
| center | C23×C4 | C24 |
| commutator, Frattini, cosocle | C2 | C28 |
| socle | C24 | C25 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C2 ▷ C1 | |
425290 conjugacy classes of subgroups.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 8 generators on 22 points:
(1,2);(3,4);(5,6);(7,8)(9,15)(10,14)(11,13)(12,16)(17,21)(18,20)(19,22);(7,9)(8,12)(10,17)(11,16)(13,15)(14,19)(18,22)(20,21);(9,16)(12,15)(17,22)(19,21);(8,13)(12,15)(14,20)(19,21);(7,10,11,18)(8,14,13,20)(9,17,16,22)(12,19,15,21)272 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10494207);