Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12316 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
| direct product | C2×#12409 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | #447 |
| commutator, Frattini, cosocle | C4×C8 | C24 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ C2×C8×Q8 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
682 conjugacy classes of subgroups.
| minimal degree | μ(G) = 50 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 4 generators on 50 points:
(1,2);(3,4,9,13)(5,14,6,17)(7,12,8,11)(10,16,18,15)(19,20,25,29)(21,30,33,48)(22,27,24,39)(23,35,38,49)(26,41,44,32)(28,34,46,37)(31,47,40,45)(36,42,50,43);(3,5,9,6)(4,10,13,18)(7,16,8,15)(11,17,12,14)(19,21)(20,26)(22,31)(23,36)(24,40)(25,33)(27,42)(28,45)(29,44)(30,35)(32,37)(34,41)(38,50)(39,43)(46,47)(48,49);(3,6,9,5)(4,10,13,18)(7,15,8,16)(11,17,12,14)(19,22,25,24)(20,27,29,39)(21,31,33,40)(23,34,38,37)(26,42,44,43)(28,35,46,49)(30,47,48,45)(32,36,41,50)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 396318);