Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8491 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #379 |
| commutator | C42 | C22×C8 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | #1779 |
| prime radical | #3480 | C4 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
553 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 48 points:
(1,2)(3,10,12,7)(4,6)(5,9)(8,13)(11,15,16,14)(17,18,22,29,24,27,40,44,23,30,39,47,21,28,37,45)(19,31,33,46,32,25,38,41,34,26,42,43,20,35,36,48);(2,6)(4,13)(7,14)(8,9)(10,15)(11,16)(17,19)(18,25)(20,21)(22,33)(23,34)(24,32)(26,27)(28,31)(29,41)(30,35)(36,37)(38,40)(39,42)(43,44)(45,46)(47,48);(1,3,5,12)(2,7,9,10)(4,11,13,16)(6,14,8,15)(17,20,23,32)(18,26,30,31)(19,21,34,24)(22,38,39,36)(25,27,35,28)(29,46,47,43)(33,40,42,37)(41,45,48,44)Polycyclic presentation:
2^9|a^2=f,c^2=g,d^2=g,e^2=i,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=egi,d^b=dg,e^b=ei,d^c=di,e^c=ei,f^c=fg68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 36908);