Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #560 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C4 · Schur multiplier (H2) = C213 |
| direct product | C4×#3531 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C42 | C24 |
| commutator | C22 | C25×C4 |
| Frattini, socle, cosocle | C23 | C26 |
| prime radical | C2×#3531 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
6552 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 36 points:
(1,2,3,4);(5,6)(7,13)(8,12)(9,11)(10,14)(15,19)(16,18)(17,20)(21,22)(23,26)(24,31)(25,28)(27,33)(29,36)(30,35)(32,34);(5,7)(6,10)(8,15)(9,14)(11,13)(12,17)(16,20)(18,19)(21,23,25,30)(22,26,28,35)(24,32,33,29)(27,36,31,34);(7,14)(10,13)(15,20)(17,19)(21,24,25,33)(22,27,28,31)(23,29,30,32)(26,34,35,36);(6,11)(7,14)(12,18)(15,20)(22,28)(26,35)(27,31)(34,36);(5,8,9,16)(6,12,11,18)(7,15,14,20)(10,17,13,19)152 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7533134);