Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #47826 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
| direct product | C2×#52055 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C25 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
6678 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);(4,6)(7,8,12,16)(9,10,14,13)(11,15,19,22)(17,18,21,20)(23,24,28,31)(25,27,33,29)(26,30,36,38)(32,35,34,37);(9,14)(10,13)(11,19)(15,22)(23,25,28,33)(24,29,31,27)(26,32,36,34)(30,37,38,35);(3,4)(5,6)(7,9)(8,13)(10,16)(11,17)(12,14)(15,20)(18,22)(19,21)(23,25)(24,29)(26,34)(27,31)(28,33)(30,35)(32,36)(37,38);(7,10,12,13)(8,14,16,9)(11,18,19,20)(15,21,22,17)(23,26)(24,30)(25,32)(27,35)(28,36)(29,37)(31,38)(33,34);(7,11,12,19)(8,15,16,22)(9,17,14,21)(10,18,13,20)(23,27,28,29)(24,25,31,33)(26,35,36,37)(30,32,38,34)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7615329);