Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19456 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
| direct product | C2×#19487 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#121 |
| commutator, Frattini, cosocle | C42 | C25 |
| prime radical | C22×#41 | D4 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1640 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 34 points:
(1,2);(3,4,7,10)(5,11,12,18)(6,14,15,9)(8,16,17,13)(19,20,24,28)(21,25,30,34)(22,27,23,32)(26,31,33,29);(3,5,6,13,7,12,15,16)(4,8,9,11,10,17,14,18)(19,21)(20,25)(22,31)(23,29)(24,30)(26,32)(27,33)(28,34);(19,21)(20,26)(22,29)(23,31)(24,30)(25,32)(27,34)(28,33);(5,12)(8,17)(11,18)(13,16)(19,22,24,23)(20,27,28,32)(21,29,30,31)(25,26,34,33)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6261753);