Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33607 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22×D8 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| socle | C22 | C2×C4×D8 |
| prime radical | #33600 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×M16 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
652 conjugacy classes of subgroups.
| minimal degree | μ(G) = 68 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 68 points:
(1,2)(3,4)(5,6,11,17,12,19,33,18)(7,20,25,48,26,37,55,49)(8,14,29,42,30,44,59,43)(9,15,10,32,21,16,34,46)(13,35,38,56,40,23,39,24)(22,47,52,65,53,63,68,66)(27,41,57,64,31,45,28,58)(36,50,60,67,62,54,61,51);(5,7,21,35,11,25,34,56,12,26,9,23,33,55,10,24)(6,13,32,49,18,39,15,37,19,40,46,48,17,38,16,20)(8,22,27,50,29,52,57,67,30,53,31,54,59,68,28,51)(14,36,58,66,43,61,45,63,44,62,64,65,42,60,41,47);(2,4)(5,8)(6,14)(7,22)(9,31)(10,28)(11,29)(12,30)(13,36)(15,45)(16,41)(17,42)(18,43)(19,44)(20,47)(21,27)(23,54)(24,51)(25,52)(26,53)(32,58)(33,59)(34,57)(35,50)(37,63)(38,60)(39,61)(40,62)(46,64)(48,65)(49,66)(55,68)(56,67);(5,9,11,10,12,21,33,34)(6,15,17,32,19,16,18,46)(7,23,25,24,26,35,55,56)(8,27,29,57,30,31,59,28)(13,37,38,49,40,20,39,48)(14,41,42,64,44,45,43,58)(22,50,52,67,53,54,68,51)(36,47,60,65,62,63,61,66)Polycyclic presentation:
2^9|a^2=g,b^2=eh,d^2=g,e^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,e^a=eh,h^a=hi,d^c=di104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 403076);