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×#12410 |
| 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 | |
698 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)(19,20,24,28)(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);(3,5,9,6)(4,10,13,18)(7,16,8,15)(11,17,12,14)(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);(3,6,9,5)(4,10,13,18)(7,15,8,16)(11,17,12,14)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 396316);