Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #40484 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#42243 |
| 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 | |
5012 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);(5,9)(8,10)(11,12)(13,17)(14,16)(15,18)(19,20,24,22)(21,25,27,29)(23,26,31,30)(28,32,33,34);(3,4,7,6)(5,8,9,10)(11,12,14,16)(13,15,18,17)(21,27)(23,31)(25,29)(26,30);(3,5,7,9)(4,8,6,10)(11,12)(13,15)(14,16)(17,18)(19,21)(20,25)(22,29)(23,28)(24,27)(26,32)(30,34)(31,33);(11,13)(12,15)(14,18)(16,17)(19,22,24,20)(21,25,27,29)(23,30,31,26)(28,32,33,34);(3,6,7,4)(5,10,9,8)(19,23)(20,26)(21,28)(22,30)(24,31)(25,32)(27,33)(29,34)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7547794);