Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #1355 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C16 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C8 | #329 |
| commutator | C42 | C2×C16 |
| Frattini, cosocle | C2×#346 | C22 |
| socle | C2 | #7128 |
| prime radical | C23 | #349 |
| LCS | G ▷ C42 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8 ◁ C22×C8 ◁ C2×#346 ◁ G | |
| LECS | G ▷ C2×#346 ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
130 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 2 generators on 64 points:
(1,2,4,10,7,12,23,37,5,11,22,36,25,39,50,61,6,13,21,38,26,41,49,63,24,9,32,33,52,34,58,59)(3,14,16,35,18,42,45,60,19,8,20,28,47,29,48,56,15,30,43,55,27,53,54,62,17,31,44,57,46,40,51,64);(1,3,6,15)(2,8)(4,20,21,44)(5,19,24,17)(7,27,26,18)(9,14)(10,35)(11,30)(12,40)(13,31)(16,22,43,32)(23,51,49,48)(25,46,52,47)(28,36)(29,41)(33,57)(34,53)(37,62)(38,55)(39,42)(45,58,54,50)(56,59)(60,63)(61,64)Polycyclic presentation:
2^9|a^2=d,b^2=f,c^2=fi,d^2=g,e^2=i,g^2=h,h^2=i,b^a=bc,c^a=ce,f^a=fi,c^b=cf,d^b=def,e^b=ei,g^b=gi,d^c=di80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1164);