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×#51588 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
6664 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)(23,24,28,26)(25,30,33,29)(27,34,36,31)(32,37,35,38);(8,12)(9,15)(10,18)(20,22)(23,25,28,33)(24,29,26,30)(27,35,36,32)(31,38,34,37);(3,4,5,6)(7,8)(9,16)(10,14)(11,12)(13,15)(17,22)(18,19)(20,21)(23,25)(24,30)(26,29)(27,32)(28,33)(31,38)(34,37)(35,36);(7,9)(8,13)(10,17)(11,15)(12,16)(14,20)(18,21)(19,22)(23,26,28,24)(25,30,33,29)(27,34,36,31)(32,38,35,37);(7,10,11,18)(8,14,12,19)(9,17,15,21)(13,20,16,22)(23,27,28,36)(24,31,26,34)(25,32,33,35)(29,37,30,38)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7614819);