Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33618 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
| direct product | C2×#33619 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#466 |
| commutator, Frattini, cosocle | C2×C8 | C25 |
| prime radical | C2×#3067 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C22×#131 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
1892 conjugacy classes of subgroups.
| minimal degree | μ(G) = 66 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 66 points:
(1,2);(3,4,10,16)(5,17,20,43)(6,24,27,41)(7,28,30,39)(8,15,9,14)(11,32,34,21)(12,31,38,26)(13,29,40,18)(19,48,49,65)(22,45,23,44)(25,56,57,37)(33,60,61,66)(35,52,36,46)(42,53,64,51)(47,55,58,63)(50,54,59,62);(3,5,18,46,9,23,30,32,10,20,29,52,8,22,7,21)(4,11,28,45,15,36,40,43,16,34,39,44,14,35,13,17)(6,19,47,66,26,51,57,59,27,49,58,60,31,53,25,50)(12,33,55,48,24,54,56,64,38,61,63,65,41,62,37,42);(3,6)(4,12)(5,19)(7,25)(8,31)(9,26)(10,27)(11,33)(13,37)(14,41)(15,24)(16,38)(17,42)(18,47)(20,49)(21,50)(22,53)(23,51)(28,55)(29,58)(30,57)(32,59)(34,61)(35,62)(36,54)(39,63)(40,56)(43,64)(44,65)(45,48)(46,66)(52,60);(5,20)(11,34)(17,43)(19,49)(21,32)(22,23)(33,61)(35,36)(42,64)(44,45)(46,52)(48,65)(50,59)(51,53)(54,62)(60,66)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7530186);