Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #511495 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #32613 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | C2×#509 | C22 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ #2960 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
431 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,8,12,6,11,5,25)(3,14,17,9,19,40,16,34)(4,20,22,37,24,10,32,39)(7,28,31,59,27,13,26,41)(15,44,45,62,47,61,46,33)(18,49,48,63,51,42,29,35)(21,54,50,36,56,64,55,52)(23,38,57,60,58,53,30,43);(1,3)(2,9)(4,21)(5,16)(6,19)(7,29)(8,17)(10,36)(11,34)(12,40)(13,42)(14,25)(15,23)(18,31)(20,52)(22,50)(24,56)(26,51)(27,48)(28,49)(30,46)(32,55)(33,38)(35,41)(37,54)(39,64)(43,61)(44,60)(45,57)(47,58)(53,62)(59,63);(1,4,5,22,6,24,8,32)(2,10,12,37,11,20,25,39)(3,15,17,46,19,47,16,45)(7,30,26,23,27,57,31,58)(9,33,14,44,34,62,40,61)(13,43,41,53,28,60,59,38)(18,50,48,21,51,55,29,56)(35,64,42,52,63,54,49,36);(2,11)(3,16,19,17)(9,14,34,40)(10,20)(12,25)(13,28)(15,45,47,46)(18,29,51,48)(21,50,56,55)(33,44,62,61)(35,42,63,49)(36,64,52,54)(37,39)(38,53)(41,59)(43,60)Polycyclic presentation:
2^9|a^2=ef,c^2=e,d^2=efhi,e^2=i,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=defh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gh,h^b=hi,f^c=fi,g^c=ghi,h^c=hi44 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 418692);