Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6370843 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | C22×#117 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| socle | C2 | C22×#382 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ #3428 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2221 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)(3,13,17,8)(4,18,21,45)(5,10)(6,12)(7,11)(9,20,29,48)(14,36)(15,34,40,26)(16,35,42,27)(19,44,46,62)(22,30)(23,31)(24,32)(25,38)(28,47,52,63)(33,54)(37,57)(39,43)(41,56,60,51)(49,53)(50,59)(55,64)(58,61);(1,3,6,16,7,17,24,42)(2,8,12,27,11,13,32,35)(4,14,20,38,21,39,48,54)(5,15,23,41,22,40,49,60)(9,25,18,43,29,33,45,36)(10,26,31,51,30,34,53,56)(19,37,47,59,46,58,63,64)(28,50,44,61,52,55,62,57);(1,4)(2,9,11,29)(3,14)(5,19)(6,20)(7,21)(8,25,13,33)(10,28,30,52)(12,18,32,45)(15,37)(16,38)(17,39)(22,46)(23,47)(24,48)(26,50,34,55)(27,43,35,36)(31,44,53,62)(40,58)(41,59)(42,54)(49,63)(51,61,56,57)(60,64);(1,5)(2,10)(3,15)(4,19)(6,23)(7,22)(8,26)(9,28)(11,30)(12,31)(13,34)(14,37)(16,41)(17,40)(18,44)(20,47)(21,46)(24,49)(25,50)(27,51)(29,52)(32,53)(33,55)(35,56)(36,57)(38,59)(39,58)(42,60)(43,61)(45,62)(48,63)(54,64);(5,22)(10,30)(15,40)(19,46)(23,49)(26,34)(28,52)(31,53)(37,58)(41,60)(44,62)(47,63)(50,55)(51,56)(57,61)(59,64)Polycyclic presentation:
2^9|a^2=h,b^2=g,c^2=f,g^2=i,b^a=bf,c^a=cg,f^a=fi,h^b=hi,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, 6278232);