Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32870 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C43 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2826 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14645 |
| prime radical | C22×#135 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
788 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)(3,14,19,9)(4,20,25,42)(5,11)(6,13)(7,31)(8,12)(10,23,37,24)(15,46)(16,45,51,33)(17,41,18,34)(21,44)(22,53,27,58)(26,38,52,60)(28,40)(29,39)(30,43)(32,48)(35,49)(36,56,62,55)(47,64)(50,61)(54,63)(57,59);(1,3)(2,9)(4,21)(5,26)(6,18)(7,17)(8,19)(10,35)(11,38)(12,14)(13,41)(15,24)(16,30)(20,32)(22,47)(23,49)(25,48)(27,50)(28,52)(29,51)(31,34)(33,39)(36,59)(37,46)(40,60)(42,44)(43,45)(53,64)(54,56)(55,57)(58,61)(62,63);(1,4)(2,10)(3,15)(5,27)(6,24)(7,23)(8,25)(9,32)(11,36)(12,37)(13,42)(14,44)(16,50)(17,21)(18,48)(19,49)(20,31)(22,28)(26,57)(29,55)(30,56)(33,59)(34,46)(35,41)(38,61)(39,53)(40,62)(43,58)(45,63)(47,51)(52,54)(60,64);(1,5,7,29,8,28,6,30)(2,11,31,39,12,40,13,43)(3,16,18,52,19,51,17,26)(4,22,24,55,25,27,23,56)(9,33,41,38,14,45,34,60)(10,36,42,58,37,62,20,53)(15,47,21,54,49,50,48,57)(32,59,46,64,44,63,35,61)Polycyclic presentation:
2^9|a^2=h,d^2=eg,f^2=i,g^2=i,b^a=be,c^a=cf,e^a=ei,g^a=gi,c^b=cg,d^b=def,f^b=fi,g^b=gi,h^b=hi,d^c=de,f^c=fi,g^c=gi,h^c=hi,e^d=ei,f^d=fi,g^d=gi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 298793);