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×#50998 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C22×#121 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
5408 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);(4,6)(8,9)(11,12)(13,17)(14,16)(15,18)(19,20,24,27)(21,28,29,25)(22,26,32,23)(30,34,33,31);(3,4,7,6)(5,9,10,8)(11,12,14,16)(13,15,18,17)(21,29)(22,32)(23,26)(25,28);(11,13)(12,15)(14,18)(16,17)(19,21)(20,25)(22,33)(23,31)(24,29)(26,34)(27,28)(30,32);(3,5,7,10)(4,8,6,9)(11,12)(13,15)(14,16)(17,18)(19,22)(20,26)(21,30)(23,27)(24,32)(25,31)(28,34)(29,33);(3,6,7,4)(5,9,10,8)(11,13)(12,15)(14,18)(16,17)(19,23,24,26)(20,22,27,32)(21,31,29,34)(25,33,28,30)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7613741);