Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #360529 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #13877 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×2−1+4 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2091 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)(3,13,17,8)(4,18)(5,23,25,10)(6,26)(7,11)(9,21)(12,28)(14,38,19,45)(15,30)(16,31,27,35)(20,48,50,43)(22,44)(24,34,55,53)(29,40,36,47)(32,49,52,64)(33,51)(37,42)(39,61)(41,58)(46,60)(54,57)(56,62)(59,63);(1,3)(2,8)(4,19)(5,15)(6,27)(7,17)(9,29)(10,30)(11,13)(12,35)(14,22)(16,28)(18,38)(20,46)(21,40)(23,37)(24,54)(25,42)(26,31)(32,56)(33,36)(34,58)(39,50)(41,55)(43,60)(44,45)(47,51)(48,61)(49,62)(52,59)(53,57)(63,64);(1,4,6,21,7,22,28,51)(2,9,12,18,11,33,26,44)(3,14,16,40,17,19,27,47)(5,20,24,49,25,50,55,64)(8,29,31,45,13,36,35,38)(10,32,34,43,23,52,53,48)(15,39,41,62,42,46,54,63)(30,56,57,61,37,59,58,60);(1,5,7,25)(2,10,11,23)(3,15,17,42)(4,20,22,50)(6,24,28,55)(8,30,13,37)(9,32,33,52)(12,34,26,53)(14,39,19,46)(16,41,27,54)(18,43,44,48)(21,49,51,64)(29,56,36,59)(31,57,35,58)(38,60,45,61)(40,62,47,63)Polycyclic presentation:
2^9|a^2=h,c^2=f,d^2=i,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,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, 267726);