Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33854 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | D4×D8 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| socle | C2 | #22995 |
| prime radical | #33594 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ #1772 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
605 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)(3,15,19,42)(4,21)(5,27)(6,12)(7,31)(8,14)(9,18,37,20)(10,24)(11,29)(13,30)(16,46,50,63)(17,51,52,36)(22,40)(23,38)(25,59)(26,56)(28,35,53,45)(32,43)(33,41)(34,47,61,64)(39,58)(44,57,62,49)(48,60,54,55);(1,3,6,20,8,19,30,18)(2,9,12,42,14,37,13,15)(4,16,23,54,26,50,25,48)(5,28,7,17,29,53,33,52)(10,34,39,62,40,61,43,44)(11,36,41,35,27,51,31,45)(21,55,38,63,56,60,59,46)(22,57,32,47,24,49,58,64);(1,4)(2,10)(3,16)(5,22)(6,23)(7,32)(8,26)(9,34)(11,38)(12,39)(13,43)(14,40)(15,44)(17,47)(18,48)(19,50)(20,54)(21,31)(24,29)(25,30)(27,59)(28,57)(33,58)(35,60)(36,63)(37,61)(41,56)(42,62)(45,55)(46,51)(49,53)(52,64);(1,5,6,7,8,29,30,33)(2,11,12,41,14,27,13,31)(3,17,18,28,19,52,20,53)(4,22,23,32,26,24,25,58)(9,35,15,36,37,45,42,51)(10,38,39,56,40,59,43,21)(16,47,48,57,50,64,54,49)(34,60,44,63,61,55,62,46)Polycyclic presentation:
2^9|a^2=g,b^2=eg,d^2=e,e^2=i,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=di,f^a=fh,h^a=hi,d^b=de,e^b=ei,g^b=gi,f^c=fh,h^c=hi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 405511);