Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #25440 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28×C4 |
| direct product | C2×#28505 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | D42 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C22×#135 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3046 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);(3,4,5,6)(7,8)(9,18)(10,23)(11,13)(12,17)(14,22)(15,25)(16,24)(19,27)(20,34)(21,30)(26,28)(29,33)(31,38)(32,35)(36,37);(8,13,17,16)(9,19,22,21)(15,28,31,23)(20,33,36,35);(4,6)(7,9)(8,14)(10,20)(11,21)(12,22)(13,27)(15,29)(16,30)(17,18)(19,24)(23,37)(25,35)(26,36)(28,34)(31,32)(33,38);(7,10,24,38,12,26,11,25)(8,15,13,28,17,31,16,23)(9,20,19,33,22,36,21,35)(14,29,27,34,18,32,30,37)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6372869);