Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1913 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C211 |
| direct product | C22×#1915 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #386 |
| commutator | C23 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
6074 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 36 points:
(1,2);(3,4);(5,6,9,14,10,13,15,28)(7,11,18,25,17,26,22,29)(8,19,21,30,20,12,23,27)(16,24,32,35,31,34,33,36);(5,7,15,18,10,17,9,22)(6,11,14,29,13,26,28,25)(8,16,21,33,20,31,23,32)(12,24,30,35,19,34,27,36);(5,8)(6,12)(7,16)(9,23)(10,20)(11,24)(13,19)(14,30)(15,21)(17,31)(18,33)(22,32)(25,36)(26,34)(27,28)(29,35)116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6278750);