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×#12386 |
| 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 | |
714 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);(3,4,7,10)(5,11,12,18)(6,14,15,9)(8,16,17,13)(19,20,24,28)(21,29,32,48)(22,33,37,47)(23,38,35,27)(25,40,42,30)(26,43,45,36)(31,49,39,44)(34,46,50,41);(6,15)(9,14)(11,18)(13,16)(19,21,24,32)(20,25,28,42)(22,34,37,50)(23,39,35,31)(26,44,45,49)(27,46,38,41)(29,47,48,33)(30,43,40,36);(3,5,7,12)(4,8,10,17)(6,16,15,13)(9,18,14,11)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 396405);