Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #36702 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#37075 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C2×#392 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
5018 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);(3,4,6,8)(5,7,9,10)(11,12,14,15)(13,17,18,16)(22,29)(26,31)(28,33)(32,34);(12,15)(16,17)(20,24)(21,27)(26,31)(28,33);(3,4)(5,7)(6,8)(9,10)(11,12)(13,16)(14,15)(17,18)(19,20,23,24)(21,25,27,30)(22,26,29,31)(28,32,33,34);(11,13,14,18)(12,16,15,17)(19,21)(20,25)(22,28)(23,27)(24,30)(26,32)(29,33)(31,34);(3,5,6,9)(4,7,8,10)(19,22)(20,26)(21,28)(23,29)(24,31)(25,32)(27,33)(30,34)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7541501);