Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #389036 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14533 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | #3449 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1201 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)(3,15)(4,21,26,37)(5,11,29,42)(6,13)(7,12)(8,14)(9,18)(10,25,38,24)(16,45,48,31)(17,44,49,39)(19,34)(20,33)(22,35,47,43)(23,53)(27,58,50,32)(28,41,30,40)(36,55)(46,59)(51,57)(52,63)(54,61)(56,62)(60,64);(1,3,8,20)(2,9,14,34)(4,22,26,47)(5,27,29,50)(6,19,7,18)(10,35,38,43)(11,39,42,44)(12,33,13,15)(16,24,48,25)(17,30,49,28)(21,45,37,31)(23,54,57,46)(32,41,58,40)(36,61,62,59)(51,63,53,60)(52,56,64,55);(1,4,6,24,8,26,7,25)(2,10,13,37,14,38,12,21)(3,16,18,47,20,48,19,22)(5,23,30,56,29,57,28,55)(9,31,33,35,34,45,15,43)(11,36,40,53,42,62,41,51)(17,46,50,64,49,54,27,52)(32,59,39,60,58,61,44,63);(1,5)(2,11)(3,17)(4,23)(6,30)(7,28)(8,29)(9,32)(10,36)(12,41)(13,40)(14,42)(15,44)(16,46)(18,50)(19,27)(20,49)(21,51)(22,52)(24,56)(25,55)(26,57)(31,59)(33,39)(34,58)(35,60)(37,53)(38,62)(43,63)(45,61)(47,64)(48,54)Polycyclic presentation:
2^9|a^2=h,b^2=i,c^2=fh,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,g^a=gi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gi,f^c=fi,h^c=hi,e^d=ei,f^d=fi,h^d=hi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 296233);