Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33691 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #3109 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| socle | C2 | #21614 |
| prime radical | C2×#135 | C23 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ #1726 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
523 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,6,13,8,14,27,12)(3,15,18,9,20,34,50,35)(4,21,23,39,26,57,25,56)(5,11,28,40,29,41,7,30)(10,22,37,58,38,31,36,24)(16,44,46,55,49,62,48,64)(17,43,51,33,52,63,19,53)(32,59,60,54,61,47,42,45);(1,3,6,18,8,20,27,50)(2,9,12,15,14,35,13,34)(4,16,23,46,26,49,25,48)(5,17,28,51,29,52,7,19)(10,32,36,42,38,61,37,60)(11,33,30,43,41,53,40,63)(21,55,56,44,57,64,39,62)(22,45,58,59,31,54,24,47);(1,4)(2,10)(3,16)(5,24)(6,23)(7,31)(8,26)(9,32)(11,39)(12,36)(13,37)(14,38)(15,42)(17,47)(18,46)(19,54)(20,49)(21,30)(22,28)(25,27)(29,58)(33,62)(34,60)(35,61)(40,57)(41,56)(43,55)(44,53)(45,51)(48,50)(52,59)(63,64);(1,5,27,7,8,29,6,28)(2,11,12,30,14,41,13,40)(3,17,50,19,20,52,18,51)(4,22,23,58,26,31,25,24)(9,33,15,43,35,53,34,63)(10,21,37,39,38,57,36,56)(16,45,46,59,49,54,48,47)(32,55,60,62,61,64,42,44)Polycyclic presentation:
2^9|a^2=g,b^2=e,d^2=egh,e^2=i,f^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fh,h^a=hi,d^c=dh,f^c=fh,h^c=hi,f^d=fi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 403211);