Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1812 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C23×C42 |
| direct product | C4×#1833 |
| subgroup | G/H | |
|---|---|---|
| center | C42 | #117 |
| commutator | C2×C4 | C22×C42 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C2×#382 |
| prime radical | C24 | C2×#41 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C42 ◁ C4×#117 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
972 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2,3,4);(5,6)(7,15,18,11)(8,19,21,34)(9,14)(10,13)(12,20,25,35)(16,29)(17,28,32,24)(22,26)(23,30)(27,36)(31,33);(5,7,9,17,10,18,22,32)(6,11,14,24,13,15,26,28)(8,16,20,30,21,31,35,36)(12,23,19,33,25,27,34,29);(5,8)(6,12,13,25)(7,16)(9,20)(10,21)(11,23,15,27)(14,19,26,34)(17,30)(18,31)(22,35)(24,33,28,29)(32,36)128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 262230);