Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #360526 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #13877 |
| 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 | |
2085 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,6,12,7,11,27,35)(3,13,17,36,16,8,25,30)(4,9,21,33,20,32,50,57)(5,22,23,37,24,10,26,34)(14,28,41,53,18,38,46,58)(15,29,43,54,42,39,44,55)(19,47,49,59,48,31,51,56)(40,60,61,64,45,52,62,63);(1,3)(2,8)(4,18)(5,15)(6,25)(7,16)(9,28)(10,29)(11,13)(12,36)(14,20)(17,27)(19,45)(21,41)(22,39)(23,44)(24,42)(26,43)(30,35)(31,52)(32,38)(33,58)(34,55)(37,54)(40,48)(46,50)(47,60)(49,61)(51,62)(53,57)(56,64)(59,63);(1,4)(2,9)(3,14,16,18)(5,19)(6,21)(7,20)(8,28,13,38)(10,31)(11,32)(12,33)(15,40,42,45)(17,41,25,46)(22,47)(23,49)(24,48)(26,51)(27,50)(29,52,39,60)(30,53,36,58)(34,56)(35,57)(37,59)(43,61,44,62)(54,63,55,64);(1,5,7,24)(2,10,11,22)(3,15,16,42)(4,19,20,48)(6,26,27,23)(8,29,13,39)(9,31,32,47)(12,37,35,34)(14,40,18,45)(17,44,25,43)(21,51,50,49)(28,52,38,60)(30,55,36,54)(33,59,57,56)(41,62,46,61)(53,64,58,63)Polycyclic presentation:
2^9|a^2=h,c^2=f,d^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,g^a=gi,c^b=cg,f^b=fi,h^b=hi,g^c=gi,h^d=hi77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 267724);