Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24986 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C8 | #555 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| socle | C2 | #23274 |
| prime radical | #575 | C2×C4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8 ◁ C22×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
402 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,8,14,6,15,17,27,7,13,32,19,31,12,43,34)(3,16,22,46,20,9,33,36,21,29,44,57,53,5,11,30)(4,10,25,39,24,40,47,48,26,41,56,52,55,38,63,60)(18,45,50,64,49,35,59,61,51,23,37,54,62,28,42,58);(1,3,17,33,31,53,8,22,7,21,43,11,6,20,32,44)(2,9,34,46,12,16,19,30,13,5,27,57,15,29,14,36)(4,18,47,59,55,62,25,50,26,51,63,42,24,49,56,37)(10,35,60,64,38,45,52,58,41,28,48,54,40,23,39,61);(1,4,6,24,7,26,31,55)(2,10,12,38,13,41,15,40)(3,18,20,49,21,51,53,62)(5,28,29,23,9,35,16,45)(8,25,17,47,32,56,43,63)(11,42,22,50,33,59,44,37)(14,39,34,60,19,52,27,48)(30,58,57,54,36,61,46,64);(1,5,7,9)(2,11,13,33)(3,19,21,14)(4,23)(6,16,31,29)(8,30,32,36)(10,37)(12,44,15,22)(17,46,43,57)(18,48)(20,34,53,27)(24,28)(25,54)(26,45)(35,55)(38,59)(39,62)(40,42)(41,50)(47,58)(49,52)(51,60)(56,64)(61,63)Polycyclic presentation:
2^9|a^2=g,b^2=eg,c^2=e,d^2=f,e^2=i,g^2=h,h^2=i,b^a=be,c^a=cf,e^a=ei,d^b=de,d^c=dh,f^c=fi,e^d=ei80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 405867);