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×#51325 |
| 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 | |
6624 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)(23,24,29,34)(25,35,36,30)(26,31,33,28)(27,32,37,38);(8,12)(9,15)(10,18)(20,22)(23,25,29,36)(24,30,34,35)(26,37,33,27)(28,38,31,32);(3,4)(5,6)(7,8,11,12)(9,16,15,13)(10,14,18,19)(17,22,21,20)(23,26,29,33)(24,31,34,28)(25,27,36,37)(30,38,35,32);(7,9,11,15)(8,13,12,16)(10,17,18,21)(14,20,19,22)(23,27)(24,32)(25,26)(28,30)(29,37)(31,35)(33,36)(34,38);(7,10)(8,14)(9,17)(11,18)(12,19)(13,20)(15,21)(16,22)(23,28,29,31)(24,33,34,26)(25,32,36,38)(27,30,37,35)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7614108);