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×#51206 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
6446 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);(4,6)(7,8,12,10)(9,13,16,17)(11,14,20,18)(15,21,19,22)(23,24,28,33)(25,30,35,29)(26,31,36,37)(27,34,38,32);(7,9,12,16)(8,13,10,17)(11,19,20,15)(14,22,18,21)(23,25,28,35)(24,29,33,30)(26,34,36,32)(27,37,38,31);(3,4,5,6)(7,9)(8,13)(10,17)(11,15)(12,16)(14,21)(18,22)(19,20)(23,25)(24,30)(26,32)(27,31)(28,35)(29,33)(34,36)(37,38);(7,10,12,8)(9,13,16,17)(11,18,20,14)(15,21,19,22)(23,26)(24,31)(25,34)(27,29)(28,36)(30,38)(32,35)(33,37);(7,11)(8,14)(9,15)(10,18)(12,20)(13,21)(16,19)(17,22)(23,27,28,38)(24,32,33,34)(25,31,35,37)(26,29,36,30)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7614110);