Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8528 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
| direct product | C2×#8529 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #379 |
| commutator | C42 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| prime radical | C2×#3553 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
2074 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);(4,9)(5,12)(6,7)(11,14)(13,15)(16,17)(19,20)(21,29,32,25)(22,33,34,44)(23,28)(24,27)(26,35,41,37)(30,46)(31,43,36,40)(38,42)(39,48)(45,49)(47,50);(3,4,8,11)(5,10,13,18)(6,14,7,9)(12,17,15,16)(19,21,24,32)(20,25,27,29)(22,30,34,47)(23,36,38,31)(26,39,41,45)(28,43,42,40)(33,46,44,50)(35,49,37,48);(3,5)(4,10)(6,12)(7,15)(8,13)(9,16)(11,18)(14,17)(19,22)(20,26)(21,30)(23,37)(24,34)(25,39)(27,41)(28,44)(29,45)(31,49)(32,47)(33,42)(35,38)(36,48)(40,46)(43,50)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 95058);