Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #47826 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#51515 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C2×#2673 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
5624 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)(5,9)(6,7)(8,10)(11,12)(13,17)(14,15)(16,18)(19,20,24,27)(21,28,23,25)(22,30,32,26)(29,33,31,34);(4,7)(8,9)(12,15)(16,17)(19,21,24,23)(20,25,27,28)(22,31,32,29)(26,34,30,33);(11,13)(12,16)(14,18)(15,17)(19,22,24,32)(20,26,27,30)(21,29,23,31)(25,33,28,34);(3,4)(5,8)(6,7)(9,10)(11,12)(13,16)(14,15)(17,18)(19,23,24,21)(20,25,27,28)(22,29,32,31)(26,34,30,33);(3,5,6,10)(4,8,7,9)(11,13)(12,16)(14,18)(15,17)(19,20)(21,28)(22,26)(23,25)(24,27)(29,34)(30,32)(31,33)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7614575);