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×#56538 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C2×#2676 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
5472 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);(3,4,7,5)(6,9,10,8)(19,20)(21,22)(23,31)(24,28)(25,26)(27,32)(29,33)(30,34);(4,5)(8,9)(11,12)(13,17)(14,16)(15,18)(19,20,24,28)(21,25,26,22)(23,31,32,27)(29,30,34,33);(11,12,14,16)(13,15,18,17)(19,21)(20,25)(22,28)(23,29)(24,26)(27,33)(30,31)(32,34);(3,5,7,4)(6,9,10,8)(11,12)(13,15)(14,16)(17,18)(19,22,24,25)(20,26,28,21)(23,30,32,33)(27,34,31,29);(3,6,7,10)(4,8,5,9)(11,13)(12,15)(14,18)(16,17)(19,23)(20,27)(21,29)(22,30)(24,32)(25,33)(26,34)(28,31)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8006268);