Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9495 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
| direct product | C2×#9514 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #386 |
| commutator | C24 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C25 | C22×C4 |
| LCS | G ▷ C24 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1300 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 34 points:
(1,2);(3,4)(5,8,12,15)(6,9)(7,10)(11,14,17,18)(13,16)(19,20,23,27,24,21,25,29)(22,28,31,34,30,26,32,33);(3,5,7,12)(4,8)(6,11,13,17)(9,14)(10,15)(16,18)(19,21,23,29,24,20,25,27)(22,28,32,33,30,26,31,34);(3,6,7,13)(4,9,10,16)(5,11,12,17)(8,14,15,18)(19,22)(20,26)(21,28)(23,32)(24,30)(25,31)(27,34)(29,33)74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 109005);