Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61629 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
| direct product | C2×#61936 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C26 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#533 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
5994 conjugacy classes of subgroups.
| minimal degree | μ(G) = 50 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 50 points:
(1,2);(3,4)(5,13)(6,10)(7,11)(8,12)(9,15)(14,18)(16,17)(19,20,24,29)(21,31,34,26)(22,35,37,44)(23,30,25,28)(27,39,42,36)(32,46,48,50)(33,41,38,43)(40,49,45,47);(3,5,8,15)(4,9,12,13)(6,14,7,16)(10,17,11,18)(19,21,24,34)(20,26,29,31)(22,32,37,48)(23,38,25,33)(27,40,42,45)(28,43,30,41)(35,50,44,46)(36,49,39,47);(3,6,8,7)(4,10,12,11)(5,14,15,16)(9,17,13,18)(19,22)(20,27)(21,32)(23,36)(24,37)(25,39)(26,40)(28,35)(29,42)(30,44)(31,45)(33,47)(34,48)(38,49)(41,46)(43,50);(3,7,8,6)(4,11,12,10)(5,14,15,16)(9,17,13,18);(4,12)(9,13)(10,11)(17,18)(19,23,24,25)(20,28,29,30)(21,33,34,38)(22,36,37,39)(26,41,31,43)(27,35,42,44)(32,47,48,49)(40,46,45,50)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10485808);