Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9331 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
| direct product | C2×#9331 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #386 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#525 | C22 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1182 conjugacy classes of subgroups.
| minimal degree | μ(G) = 50 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 50 points:
(1,2);(3,4)(5,8,12,16)(6,13,14,9)(7,10)(11,15)(17,18)(19,20)(21,30)(22,34,37,44)(23,38)(24,28)(25,29)(26,32)(27,36,43,35)(31,46,48,40)(33,41)(39,42)(45,47,50,49);(3,5)(4,8,10,16)(6,11)(7,12)(9,15,13,18)(14,17)(19,21,23,32,25,33,24,39)(20,26,28,30,29,42,38,41)(22,31,36,49,37,48,35,47)(27,40,44,45,43,46,34,50);(3,6)(4,9)(5,11)(7,14)(8,15)(10,13)(12,17)(16,18)(19,22)(20,27)(21,31)(23,36)(24,35)(25,37)(26,40)(28,44)(29,43)(30,45)(32,49)(33,48)(34,38)(39,47)(41,50)(42,46)68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 106712);