Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #53390 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#58625 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C2×#2635 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
5200 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 34 points:
(1,2);(4,8)(6,10)(12,15)(13,17);(3,4,7,8)(5,10,9,6)(11,12,14,15)(13,18,17,16)(20,24)(22,30)(25,27)(28,34);(4,8)(5,9)(11,12,14,15)(13,16,17,18)(19,20,23,24)(21,27,29,25)(22,26,30,31)(28,33,34,32);(3,5,7,9)(4,6,8,10)(11,12)(13,16)(14,15)(17,18)(19,21,23,29)(20,25,24,27)(22,28,30,34)(26,32,31,33);(3,6)(4,9)(5,8)(7,10)(11,13,14,17)(12,16,15,18)(19,22)(20,26)(21,28)(23,30)(24,31)(25,32)(27,33)(29,34)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8012004);