Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #382865 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14345 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1126 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,6,11,7,12,26,37)(3,13,16,29,17,36,25,8)(4,18,20,49,21,9,30,32)(5,10,23,34,24,35,56,60)(14,40,42,27,43,38,51,47)(15,39,45,59,46,62,55,28)(19,48,22,52,50,31,54,33)(41,61,44,57,64,58,53,63);(1,3,7,17)(2,8,12,29)(4,14,21,43)(5,15,24,46)(6,25,26,16)(9,27,18,47)(10,28,35,59)(11,36,37,13)(19,41,50,64)(20,51,30,42)(22,53,54,44)(23,55,56,45)(31,57,48,63)(32,40,49,38)(33,61,52,58)(34,62,60,39);(1,4)(2,9,12,18)(3,14)(5,22,24,54)(6,20)(7,21)(8,27,29,47)(10,33)(11,32,37,49)(13,38,36,40)(15,44,46,53)(16,42)(17,43)(19,23,50,56)(25,51)(26,30)(28,58)(31,60)(34,48)(35,52)(39,63)(41,45,64,55)(57,62)(59,61);(1,5)(2,10)(3,15)(4,19,21,50)(6,23)(7,24)(8,28)(9,31,18,48)(11,34)(12,35)(13,39)(14,41,43,64)(16,45)(17,46)(20,22,30,54)(25,55)(26,56)(27,57,47,63)(29,59)(32,33,49,52)(36,62)(37,60)(38,58,40,61)(42,44,51,53)Polycyclic presentation:
2^9|a^2=e,b^2=i,c^2=h,d^2=f,e^2=i,b^a=be,c^a=cf,h^a=hi,c^b=cg,e^b=ei,d^c=de,f^c=fi,g^d=gi,h^d=hi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 290062);