Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #471 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C29 |
| direct product | C2×#10773 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | #121 |
| commutator | C2×C4 | C22×C42 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C23 | #380 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
4098 conjugacy classes of subgroups.
| minimal degree | μ(G) = 38 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 38 points:
(1,2);(3,4)(5,6)(7,8)(9,19)(10,24,27,32)(11,17)(12,16)(13,18)(14,21)(15,26,31,25)(20,34,37,28)(22,30)(23,29)(33,36,38,35);(4,6)(7,9,12,22,13,23,11,21)(8,14,17,29,18,30,16,19)(10,20,26,36,27,37,25,35)(15,28,32,38,31,34,24,33);(7,10)(8,15)(9,20)(11,25)(12,26)(13,27)(14,28)(16,24)(17,32)(18,31)(19,33)(21,35)(22,36)(23,37)(29,38)(30,34)128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 260549);