Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3003 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28×C4 |
| direct product | C2×#21070 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| socle | C23 | C2×C4×D4 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C24×C8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3250 conjugacy classes of subgroups.
| minimal degree | μ(G) = 38 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 38 points:
(1,2);(4,5)(7,8,12,17,13,16,11,25)(9,14,20,29,21,28,19,35)(10,22,23,15,24,31,26,30)(18,32,33,27,34,38,36,37);(3,4,6,5)(7,9,12,20,13,21,11,19)(8,14,17,29,16,28,25,35)(10,18,23,33,24,34,26,36)(15,27,31,38,30,37,22,32);(3,5,6,4)(7,10)(8,15)(9,18)(11,26)(12,23)(13,24)(14,27)(16,30)(17,31)(19,36)(20,33)(21,34)(22,25)(28,37)(29,38)(32,35);(8,16)(14,28)(15,30)(17,25)(22,31)(27,37)(29,35)(32,38)128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6278854);