Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2192 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
| direct product | C2×#12749 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #447 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| prime radical | C22×#578 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ C22×#131 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
3124 conjugacy classes of subgroups.
| minimal degree | μ(G) = 38 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 38 points:
(1,2);(3,4)(5,6)(7,8,13,18,14,10,12,17)(9,19,23,35,24,36,22,33)(11,26,29,25,30,34,28,16)(15,27,32,38,20,37,31,21);(7,9)(8,15)(10,20)(11,27)(12,22)(13,23)(14,24)(16,33)(17,31)(18,32)(19,26)(21,28)(25,35)(29,38)(30,37)(34,36);(4,6)(7,10,14,8)(9,20,24,15)(11,25,30,16)(12,17,13,18)(19,21,36,38)(22,31,23,32)(26,28,34,29)(27,35,37,33)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 400745);