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×#34387 |
| 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)(25,28)(26,31)(27,32)(29,33)(30,35)(34,36)(37,38);(8,12)(13,18)(14,19)(20,22)(24,25)(26,33)(30,32)(34,38);(7,8)(9,13)(10,14)(11,12)(15,18)(16,20)(17,19)(21,22)(23,25,28,24)(26,31,33,29)(27,32,35,30)(34,37,38,36);(7,9)(8,13)(10,16)(11,15)(12,18)(14,20)(17,21)(19,22)(23,26)(24,29)(25,31)(27,34)(28,33)(30,36)(32,37)(35,38);(7,10,11,17)(8,14,12,19)(9,16,15,21)(13,20,18,22)(23,27,28,35)(24,30,25,32)(26,34,33,38)(29,36,31,37)152 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7533153);