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×#34526 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24 | C25 |
| commutator | C22 | C25×C4 |
| Frattini, cosocle | C23 | C26 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
6406 conjugacy classes of subgroups.
| minimal degree | μ(G) = 38 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 6 generators on 38 points:
(1,2);(3,4,6,5)(7,8)(9,15)(10,12)(11,18)(13,16)(14,19)(17,20)(21,22)(23,24,27,30)(25,31,33,28)(26,29,34,36)(32,37,38,35);(8,10)(9,16)(14,18)(17,22)(23,24,27,30)(25,28,33,31)(26,29,34,36)(32,35,38,37);(3,4,6,5)(7,9)(8,13)(10,15)(11,17)(12,16)(14,20)(18,21)(19,22)(23,24)(25,28)(26,29)(27,30)(31,33)(32,35)(34,36)(37,38);(7,10,12,8)(9,15,16,13)(11,18,19,14)(17,21,22,20)(23,25)(24,28)(26,32)(27,33)(29,35)(30,31)(34,38)(36,37);(3,5,6,4)(7,11)(8,14)(9,17)(10,18)(12,19)(13,20)(15,21)(16,22)(23,26,27,34)(24,29,30,36)(25,32,33,38)(28,35,31,37)152 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7533356);