Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #45438 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#45613 |
| 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 | |
5694 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);(4,5)(8,9)(13,17)(16,18)(20,24)(21,27)(22,29)(32,34);(4,5)(6,10)(12,15)(13,17)(19,20,23,24)(21,25,27,31)(22,30,29,26)(28,34,33,32);(3,4,7,5)(6,8,10,9)(11,12)(13,16)(14,15)(17,18)(19,20)(21,25)(22,26)(23,24)(27,31)(28,32)(29,30)(33,34);(3,5,7,4)(6,9,10,8)(11,13)(12,16)(14,17)(15,18)(19,21)(20,25)(22,28)(23,27)(24,31)(26,32)(29,33)(30,34);(3,6)(4,8)(5,9)(7,10)(11,12)(13,16)(14,15)(17,18)(19,22,23,29)(20,26,24,30)(21,28,27,33)(25,32,31,34)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7609439);