Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8516 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
| direct product | C2×#8527 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #379 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| prime radical | C25 | C22⋊C4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
2140 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);(3,4,7,12,8,11,6,10)(5,13,14,18,15,9,16,17)(19,20)(21,29,32,41)(22,33,35,26)(23,27)(24,28)(25,31,39,36)(30,44)(34,48,50,40)(37,42)(38,45)(43,46)(47,49);(3,5,8,15)(4,9,11,13)(6,14,7,16)(10,17,12,18)(19,21)(20,25)(22,30)(23,36)(24,32)(26,38)(27,41)(28,39)(29,42)(31,37)(33,47)(34,49)(35,46)(40,44)(43,48)(45,50);(3,6,8,7)(4,10,11,12)(5,14,15,16)(9,17,13,18)(19,22,24,35)(20,26,28,33)(21,30,32,46)(23,34,37,50)(25,38,39,47)(27,40,42,48)(29,43,41,44)(31,45,36,49)68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 95001);