Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #498800 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | C2×#469 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| socle | C2 | #23697 |
| prime radical | #13309 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C8 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
564 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,6,12,28,42,26,39,8,13,25,40,7,14,5,11)(3,9,17,32,50,60,16,31,19,33,51,61,18,34,24,38)(4,20,21,43,29,57,56,36,22,54,55,37,23,41,27,10)(15,44,47,62,52,63,64,35,46,53,45,58,48,59,49,30);(1,3)(2,9)(4,15)(5,24)(6,17)(7,18)(8,19)(10,35)(11,38)(12,32)(13,33)(14,34)(16,26)(20,53)(21,47)(22,46)(23,48)(25,51)(27,49)(28,50)(29,52)(30,36)(31,39)(37,62)(40,61)(41,63)(42,60)(43,58)(44,54)(45,55)(56,64)(57,59);(1,4)(2,10)(3,15)(5,21)(6,27)(7,29)(8,22)(9,30)(11,20)(12,41)(13,36)(14,43)(16,45)(17,49)(18,52)(19,46)(23,28)(24,47)(25,56)(26,55)(31,53)(32,59)(33,35)(34,62)(37,42)(38,44)(39,54)(40,57)(48,50)(51,64)(58,60)(61,63);(1,5,7,25,8,26,28,6)(2,11,14,40,13,39,42,12)(3,16,18,17,19,24,50,51)(4,21,29,56,22,55,23,27)(9,31,34,32,33,38,60,61)(10,20,43,57,36,54,37,41)(15,45,52,49,46,47,48,64)(30,53,62,59,35,44,58,63)Polycyclic presentation:
2^9|a^2=f,d^2=eh,f^2=hi,h^2=i,b^a=be,c^a=cf,g^a=gi,c^b=cg,d^b=di,e^c=ei,f^c=fh,h^c=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 406703);