Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #380263 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14262 |
| commutator | C24 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×2−1+4 | C22 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1753 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,6,12,7,11,25,35)(3,13,16,36,17,8,24,28)(4,9,20,32,19,31,45,57)(5,10,23,34,22,33,49,58)(14,29,40,55,39,26,46,52)(15,37,43,59,42,27,50,53)(18,30,44,56,21,47,48,62)(38,54,60,64,41,51,61,63);(1,3,7,17)(2,8,11,13)(4,14,19,39)(5,15,22,42)(6,24,25,16)(9,29,31,26)(10,27,33,37)(12,36,35,28)(18,38,21,41)(20,46,45,40)(23,50,49,43)(30,54,47,51)(32,52,57,55)(34,59,58,53)(44,61,48,60)(56,63,62,64);(1,4)(2,9)(3,14)(5,21,22,18)(6,20)(7,19)(8,26)(10,30,33,47)(11,31)(12,32)(13,29)(15,41,42,38)(16,40)(17,39)(23,48,49,44)(24,46)(25,45)(27,51,37,54)(28,52)(34,56,58,62)(35,57)(36,55)(43,61,50,60)(53,63,59,64);(1,5)(2,10)(3,15)(4,18)(6,23)(7,22)(8,27)(9,30)(11,33)(12,34)(13,37)(14,38)(16,43)(17,42)(19,21)(20,44)(24,50)(25,49)(26,51)(28,53)(29,54)(31,47)(32,56)(35,58)(36,59)(39,41)(40,60)(45,48)(46,61)(52,63)(55,64)(57,62)Polycyclic presentation:
2^9|a^2=h,b^2=i,c^2=f,h^2=i,b^a=be,c^a=cf,e^a=ei,g^a=gi,c^b=cg,h^b=hi,d^c=de,e^c=ei,f^d=fi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 287461);