Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #31418 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2688 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14363 |
| prime radical | #60422 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1322 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)(3,13,16,8)(4,9,20,34)(5,22)(6,25)(7,11)(10,23)(12,28)(14,29)(15,36,44,30)(17,31,26,38)(18,46)(19,33,49,50)(21,39,27,35)(24,37)(32,45)(40,42)(41,60)(43,56,53,61)(47,57)(48,62,54,59)(51,52)(55,63)(58,64);(1,3)(2,8)(4,18)(5,15)(6,26)(7,16)(9,32)(10,36)(11,13)(12,38)(14,21)(17,28)(19,47)(20,45)(22,30)(23,44)(24,53)(25,31)(27,42)(29,35)(33,58)(34,46)(37,56)(39,40)(41,48)(43,52)(49,64)(50,57)(51,61)(54,63)(55,62)(59,60);(1,4)(2,9,11,34)(3,14)(5,19)(6,27)(7,20)(8,29,13,40)(10,33,22,50)(12,39,25,35)(15,41)(16,42)(17,45)(18,26)(21,28)(23,49)(24,54)(30,55,36,60)(31,46,38,32)(37,62,51,59)(43,64)(44,63)(47,53)(48,52)(56,58,61,57);(1,5)(2,10,11,22)(3,15)(4,19)(6,24)(7,23)(8,30,13,36)(9,33,34,50)(12,37,25,51)(14,41)(16,44)(17,43)(18,47)(20,49)(21,48)(26,53)(27,54)(28,52)(29,55,40,60)(31,56,38,61)(32,57,46,58)(35,59,39,62)(42,63)(45,64)Polycyclic presentation:
2^9|a^2=h,c^2=e,d^2=e,g^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^b=de,f^b=fi,g^b=gi,h^b=hi,f^c=fi,g^c=gi,h^c=hi,f^d=fi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 290958);