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×#51335 |
| 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 | |
6464 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)(24,29)(25,31)(27,35)(34,38);(8,12)(9,15)(10,18)(20,22)(23,24,28,29)(25,32,31,26)(27,36,35,30)(33,34,37,38);(3,4)(5,6)(7,8,11,12)(9,16,15,13)(10,14,18,19)(17,22,21,20)(23,25,28,31)(24,26,29,32)(27,33,35,37)(30,34,36,38);(7,9,11,15)(8,13,12,16)(10,17,18,21)(14,20,19,22)(23,26)(24,25)(27,34)(28,32)(29,31)(30,33)(35,38)(36,37);(7,10)(8,14)(9,17)(11,18)(12,19)(13,20)(15,21)(16,22)(23,27,28,35)(24,30,29,36)(25,33,31,37)(26,34,32,38)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7614109);