Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #508805 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #30137 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | #13336 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ #1747 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
429 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,5,11)(3,14,17,8)(4,21)(6,9,22,12)(7,13,27,36)(10,23)(15,42,44,39)(16,46)(18,31,20,32)(19,40,45,29)(24,51)(25,35)(26,37)(28,33,52,38)(30,47)(34,53)(41,50)(43,56)(48,62)(49,57)(54,59)(55,60)(58,63)(61,64);(1,3)(2,8)(4,16)(5,17)(6,18)(7,15)(9,32)(10,30)(11,14)(12,31)(13,39)(19,28)(20,22)(21,46)(23,47)(24,43)(25,50)(26,49)(27,44)(29,33)(34,58)(35,41)(36,42)(37,57)(38,40)(45,52)(48,54)(51,56)(53,63)(55,64)(59,62)(60,61);(2,9)(3,15)(6,22)(7,28)(8,29)(10,21)(11,12)(13,36)(14,40)(16,43)(17,44)(18,45)(19,20)(23,25)(24,54)(27,52)(30,56)(31,42)(32,39)(34,59)(35,37)(41,61)(46,58)(47,48)(49,64)(50,63)(53,55)(57,62);(1,4,22,23,5,26,6,25)(2,10,9,21,11,35,12,37)(3,16,20,47,17,49,18,50)(7,24,52,53,27,55,28,54)(8,30,32,46,14,41,31,57)(13,34,33,51,36,59,38,60)(15,43,45,63,44,64,19,48)(29,56,42,62,40,61,39,58)Polycyclic presentation:
2^9|a^2=e,d^2=ef,f^2=i,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,c^b=cg,g^b=gh,h^b=hi,d^c=de,f^c=fi,g^c=gh,h^c=hi,e^d=ei,f^d=fi44 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 416001);