Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #483412 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | C2×#427 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C4×C8 | C24 |
| socle | C2 | C2×#1568 |
| prime radical | C24 | #108 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C2×#97 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
401 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,6,13,8,14,31,43,9,15,30,12,7,34,5,29)(3,16,20,37,22,45,19,36,18,41,32,58,21,54,52,10)(4,23,25,56,27,40,24,42,28,57,33,59,26,11,38,39)(17,46,49,35,51,61,62,55,47,64,53,60,50,44,48,63);(1,3,5,19,7,21,30,20,9,18,31,52,8,22,6,32)(2,10,12,16,14,37,29,45,15,36,13,41,34,58,43,54)(4,17,24,48,26,50,25,53,28,47,38,62,27,51,33,49)(11,35,39,44,23,55,56,46,40,60,42,61,57,63,59,64);(1,4,7,26,9,28,8,27)(2,11,14,23,15,40,34,57)(3,17,21,50,18,47,22,51)(5,24,30,25,31,38,6,33)(10,35,37,55,36,60,58,63)(12,39,29,56,13,42,43,59)(16,44,45,46,41,61,54,64)(19,48,20,53,52,62,32,49);(3,18)(10,36)(16,41)(17,47)(19,52)(20,32)(21,22)(35,60)(37,58)(44,61)(45,54)(46,64)(48,62)(49,53)(50,51)(55,63)Polycyclic presentation:
2^9|a^2=f,b^2=e,c^2=g,e^2=g,f^2=h,g^2=i,h^2=i,b^a=be,c^a=ch,e^a=eg,g^a=gi,d^b=di,f^b=fi,f^c=fi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 390608);