Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #46350 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
| direct product | C2×#47211 |
| 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 | |
6614 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)(7,8,12,15)(9,10,13,18)(11,14,20,22)(16,17,19,21)(23,24,28,33)(25,30,36,29)(26,31,32,27)(34,35,37,38);(7,9,12,13)(8,10,15,18)(11,19,20,16)(14,21,22,17)(23,25,28,36)(24,29,33,30)(26,37,32,34)(27,38,31,35);(3,4,5,6)(7,9)(8,10)(11,16)(12,13)(14,17)(15,18)(19,20)(21,22)(23,25)(24,30)(26,37)(27,35)(28,36)(29,33)(31,38)(32,34);(7,10)(8,13)(9,15)(11,17)(12,18)(14,19)(16,22)(20,21)(23,26,28,32)(24,31,33,27)(25,34,36,37)(29,38,30,35);(7,11)(8,14)(9,16)(10,17)(12,20)(13,19)(15,22)(18,21)(23,27,28,31)(24,32,33,26)(25,35,36,38)(29,37,30,34)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7608096);