Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3532 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C4 · Schur multiplier (H2) = C213 |
| direct product | C2×#34395 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator | C22 | C25×C4 |
| Frattini, cosocle | C23 | C26 |
| prime radical | C22×#121 | C22 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
7070 conjugacy classes of subgroups.
| minimal degree | μ(G) = 38 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 38 points:
(1,2);(3,4,5,6)(8,12)(9,15)(14,19)(16,21)(23,24,27,28)(25,31,32,29)(26,34,35,30)(33,36,38,37);(8,12)(13,18)(14,19)(20,22)(24,28)(25,32)(30,34)(33,38);(7,8,11,12)(9,13,15,18)(10,14,17,19)(16,20,21,22)(23,24)(25,29)(26,30)(27,28)(31,32)(33,36)(34,35)(37,38);(7,9)(8,13)(10,16)(11,15)(12,18)(14,20)(17,21)(19,22)(23,25,27,32)(24,29,28,31)(26,33,35,38)(30,36,34,37);(7,10,11,17)(8,14,12,19)(9,16,15,21)(13,20,18,22)(23,26,27,35)(24,30,28,34)(25,33,32,38)(29,36,31,37)152 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7533174);