Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #501524 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #3109 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | #3054 | C22 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
587 conjugacy classes of subgroups.
| minimal degree | μ(G) = 68 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 68 points:
(5,6,13,16,32,44,31,47,10,19,11,17,12,18,9,30)(7,14,33,45,55,65,21,53,22,40,23,38,24,39,54,68)(8,25,26,43,35,41,34,15,29,59,27,48,28,46,36,42)(20,49,60,64,57,67,56,37,52,58,50,66,51,62,61,63);(1,2)(3,4)(5,7)(6,14)(8,20)(9,21)(10,22)(11,33)(12,24)(13,23)(15,37)(16,38)(17,45)(18,39)(19,40)(25,58)(26,50)(27,60)(28,51)(29,52)(30,53)(31,54)(32,55)(34,61)(35,57)(36,56)(41,62)(42,63)(43,64)(44,65)(46,67)(47,68)(48,66)(49,59);(2,4)(5,8)(6,15)(7,20)(9,26)(10,29)(11,34)(12,35)(13,36)(14,37)(16,41)(17,46)(18,48)(19,42)(21,50)(22,52)(23,56)(24,57)(25,47)(27,31)(28,32)(30,59)(33,61)(38,62)(39,66)(40,63)(43,44)(45,67)(49,53)(51,55)(54,60)(58,68)(64,65);(5,9,12,11,10,31,32,13)(6,16,44,47,19,17,18,30)(7,21,24,33,22,54,55,23)(8,26,35,34,29,27,28,36)(14,38,65,68,40,45,39,53)(15,41,43,25,42,46,48,59)(20,50,57,61,52,60,51,56)(37,62,64,58,63,67,66,49)Polycyclic presentation:
2^9|a^2=fi,d^2=ehi,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=deh,e^a=ei,c^b=cg,f^b=fi,e^c=ei,f^c=fh,h^c=hi50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 408722);