Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33607 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
| direct product | C2×#33608 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D8 |
| commutator, Frattini, cosocle | C2×C8 | C25 |
| prime radical | C22×#41 | D4 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×#41 ◁ C22×#131 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
1922 conjugacy classes of subgroups.
| minimal degree | μ(G) = 70 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 70 points:
(1,2);(3,4)(5,6)(7,8,13,18)(9,19,24,44)(10,15,25,36)(11,28,30,56)(12,31,29,17)(14,32,35,22)(16,39,40,20)(21,41,47,63)(23,51,50,43)(26,54,52,69)(27,55,53,38)(33,57,59,48)(34,45,42,58)(37,62,61,46)(49,68,67,65)(60,66,64,70);(7,9,20,45,12,23,30,32,13,24,39,58,29,50,11,22)(8,14,28,43,17,34,40,44,18,35,56,51,31,42,16,19)(10,21,46,66,27,49,52,57,25,47,62,70,53,67,26,48)(15,33,54,65,38,60,61,63,36,59,69,68,55,64,37,41);(4,6)(7,10)(8,15)(9,21)(11,26)(12,27)(13,25)(14,33)(16,37)(17,38)(18,36)(19,41)(20,46)(22,48)(23,49)(24,47)(28,54)(29,53)(30,52)(31,55)(32,57)(34,60)(35,59)(39,62)(40,61)(42,64)(43,65)(44,63)(45,66)(50,67)(51,68)(56,69)(58,70);(10,25)(15,36)(21,47)(26,52)(27,53)(33,59)(37,61)(38,55)(41,63)(46,62)(48,57)(49,67)(54,69)(60,64)(65,68)(66,70)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7530164);