Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1619 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
| direct product | C4×#1636 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | #120 |
| commutator | C42 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| socle, prime radical | C23 | C2×#120 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C43 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
958 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2,3,4);(5,6,10,13)(7,14,16,19)(8,12,9,17)(11,18,20,15)(21,22,27,31)(23,32,24,35)(25,30,26,29)(28,34,36,33);(5,7)(6,11)(8,15)(9,18)(10,16)(12,19)(13,20)(14,17)(21,23,27,24)(22,28,31,36)(25,34,26,33)(29,35,30,32);(5,8,10,9)(6,12,13,17)(7,15,16,18)(11,19,20,14)(21,24,27,23)(22,28,31,36)(25,33,26,34)(29,35,30,32)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 116881);