Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6489190 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #35237 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #25798 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1698 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 64 points:
(1,2,7,10,8,14,6,13)(3,15,16,9,20,32,19,33)(4,11,25,34,22,39,24,37)(5,26,28,40,29,35,27,12)(17,36,42,30,45,57,21,49)(18,46,47,55,48,31,43,41)(23,52,54,60,51,58,53,38)(44,62,50,64,63,56,61,59);(1,3)(2,9)(4,21)(5,18)(6,16)(7,19)(8,20)(10,15)(11,36)(12,31)(13,32)(14,33)(17,24)(22,42)(23,50)(25,45)(26,55)(27,47)(28,43)(29,48)(30,37)(34,49)(35,41)(38,59)(39,57)(40,46)(44,54)(51,61)(52,56)(53,63)(58,62)(60,64);(2,10,14,13)(3,16,20,19)(4,22)(11,37,39,34)(12,35,40,26)(17,21,45,42)(18,43,48,47)(23,51)(24,25)(30,49)(36,57)(38,52,60,58)(44,50,63,61)(53,54)(56,62)(59,64);(1,4,6,24,8,22,7,25)(2,11,13,37,14,39,10,34)(3,17,19,21,20,45,16,42)(5,23,28,54,29,51,27,53)(9,30,15,36,33,49,32,57)(12,38,26,52,40,60,35,58)(18,44,47,50,48,63,43,61)(31,56,41,59,46,62,55,64);(1,5,7,27,8,29,6,28)(2,12,13,26,14,40,10,35)(3,18,19,47,20,48,16,43)(4,23,25,53,22,51,24,54)(9,31,32,55,33,46,15,41)(11,38,37,52,39,60,34,58)(17,44,21,50,45,63,42,61)(30,56,57,64,49,62,36,59)Polycyclic presentation:
2^9|a^2=fi,c^2=gh,d^2=h,e^2=g,f^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=dh,f^b=fi,h^b=hi,d^c=di,f^e=fi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6396387);