Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #386085 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14419 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | #60645 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1546 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,7,13,8,12,6,28)(3,14,19,40,17,9,18,32)(4,20,25,50,23,10,24,35)(5,11,27,39,26,37,29,38)(15,41,43,48,44,30,21,33)(16,31,45,57,46,36,47,56)(22,34,52,61,53,49,54,60)(42,62,51,58,63,55,64,59);(1,3)(2,9)(4,21)(5,16)(6,19)(7,18)(8,17)(10,33)(11,36)(12,14)(13,40)(15,24)(20,48)(22,51)(23,43)(25,44)(26,46)(27,47)(28,32)(29,45)(30,35)(31,37)(34,59)(38,56)(39,57)(41,50)(42,52)(49,58)(53,64)(54,63)(55,61)(60,62);(1,4,6,24,8,23,7,25)(2,10,13,35,12,20,28,50)(3,15,18,21,17,44,19,43)(5,22,27,52,26,53,29,54)(9,30,32,33,14,41,40,48)(11,34,38,60,37,49,39,61)(16,42,45,51,46,63,47,64)(31,55,56,58,36,62,57,59);(1,5)(2,11,12,37)(3,16)(4,22)(6,27)(7,29)(8,26)(9,31,14,36)(10,34,20,49)(13,38,28,39)(15,42)(17,46)(18,45)(19,47)(21,51)(23,53)(24,52)(25,54)(30,55,41,62)(32,56,40,57)(33,58,48,59)(35,60,50,61)(43,64)(44,63)Polycyclic presentation:
2^9|a^2=h,c^2=g,d^2=e,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,g^a=gi,c^b=cg,d^b=de,f^b=fi,g^b=gi,h^b=hi,f^c=fi,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, 293283);