Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1655 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
| direct product | C4×#1662 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | #120 |
| commutator | C42 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C23 | C2×#120 |
| prime radical | C2×#562 | C22 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C43 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
960 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2,3,4);(5,6,10,14)(7,15,17,20)(8,13,9,12)(11,18,19,16)(21,22,25,28)(23,29,31,36)(24,32,34,27)(26,33,35,30);(5,7)(6,11)(8,18)(9,16)(10,17)(12,20)(13,15)(14,19)(21,23,25,31)(22,26,28,35)(24,33,34,30)(27,36,32,29);(5,7)(6,11)(8,16)(9,18)(10,17)(12,15)(13,20)(14,19)(24,34)(27,32)(29,36)(30,33)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 116340);