Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11502 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
| direct product | C2×#12692 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#99 |
| commutator | C24 | C23×C4 |
| Frattini, cosocle | C22×D4 | C24 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C24 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ C23×#41 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
3388 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)(9,17)(10,14)(11,24,25,15)(12,16)(13,19,27,34)(18,31)(20,32)(21,35)(22,37,38,29)(23,30)(26,33,28,36);(7,9,12,20)(8,13,16,27)(10,21,23,18)(11,19,25,34)(14,28,30,26)(15,17,24,32)(22,36,38,33)(29,35,37,31);(4,6)(7,10,12,23)(8,14,16,30)(9,18,20,21)(11,22,25,38)(13,26,27,28)(15,29,24,37)(17,31,32,35)(19,33,34,36)68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 400551);