Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60470 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C210 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | C22×#121 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| socle | C2 | C22×#380 |
| prime radical | C2×#585 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4445 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 64 points:
(1,2)(3,14)(4,20,23,33)(5,11)(6,27)(7,12)(8,13)(9,17)(10,22,34,28)(15,29,42,54)(16,38)(18,31)(19,39)(21,48,49,57)(24,35)(25,51)(26,36)(30,44)(32,50,56,52)(37,41,47,46)(40,53,60,64)(43,59)(45,55)(58,61,63,62);(1,3,6,17,8,19,7,18)(2,9,12,14,13,31,27,39)(4,15,22,41,23,42,28,46)(5,16,25,44,24,43,26,45)(10,29,33,37,34,54,20,47)(11,30,36,38,35,55,51,59)(21,40,50,61,49,60,52,62)(32,53,57,58,56,64,48,63);(1,4)(2,10)(3,15)(5,21)(6,22)(7,28)(8,23)(9,29)(11,32)(12,33)(13,34)(14,37)(16,40)(17,41)(18,46)(19,42)(20,27)(24,49)(25,50)(26,52)(30,53)(31,54)(35,56)(36,57)(38,58)(39,47)(43,60)(44,61)(45,62)(48,51)(55,64)(59,63);(1,5)(2,11)(3,16)(4,21)(6,25)(7,26)(8,24)(9,30)(10,32)(12,36)(13,35)(14,38)(15,40)(17,44)(18,45)(19,43)(20,48)(22,50)(23,49)(27,51)(28,52)(29,53)(31,55)(33,57)(34,56)(37,58)(39,59)(41,61)(42,60)(46,62)(47,63)(54,64);(5,24)(11,35)(16,43)(21,49)(25,26)(30,55)(32,56)(36,51)(38,59)(40,60)(44,45)(48,57)(50,52)(53,64)(58,63)(61,62)Polycyclic presentation:
2^9|a^2=h,b^2=f,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^c=gi,h^c=hi,e^d=ei116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6278041);