Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7535047 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #3549 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| socle | C2 | #59041 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1726 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 64 points:
(1,2)(3,16)(4,21)(5,28)(6,14)(7,13)(8,15)(9,19)(10,26)(11,30)(12,29)(17,45)(18,40)(20,43)(22,36)(23,42)(24,37)(25,39)(27,38)(31,41)(32,35)(33,48)(34,50)(44,49)(46,61)(47,60)(51,56)(52,59)(53,58)(54,62)(55,63)(57,64);(1,3,6,19,8,20,7,32)(2,9,13,16,15,35,14,43)(4,22,25,48,27,54,26,17)(5,18,29,49,31,50,30,57)(10,36,38,45,39,62,21,33)(11,40,41,64,12,34,28,44)(23,52,55,47,24,53,56,46)(37,61,63,58,42,60,51,59);(1,4,7,26,8,27,6,25)(2,10,14,21,15,39,13,38)(3,17,19,22,20,48,32,54)(5,23,30,56,31,24,29,55)(9,33,16,36,35,45,43,62)(11,37,28,51,12,42,41,63)(18,46,49,52,50,47,57,53)(34,58,44,60,40,59,64,61);(1,5,6,29,8,31,7,30)(2,11,13,41,15,12,14,28)(3,18,19,49,20,50,32,57)(4,23,25,55,27,24,26,56)(9,34,16,44,35,40,43,64)(10,37,38,63,39,42,21,51)(17,46,22,52,48,47,54,53)(33,58,36,60,45,59,62,61);(1,5,6,29,8,31,7,30)(2,12,13,28,15,11,14,41)(3,18,19,49,20,50,32,57)(4,24,25,56,27,23,26,55)(9,34,16,44,35,40,43,64)(10,37,38,63,39,42,21,51)(17,47,22,53,48,46,54,52)(33,59,36,61,45,58,62,60)Polycyclic presentation:
2^9|b^2=f,c^2=g,d^2=f,e^2=f,f^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=dh,g^b=gi,e^c=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7442243);