Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6803848 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #39148 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #25447 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1087 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)(3,14)(4,21)(5,28)(6,11)(7,33)(8,13)(9,18)(10,26)(12,29)(15,45)(16,49)(17,35)(19,38)(20,37)(22,34)(23,39)(24,53)(25,41)(27,40)(30,58)(31,44)(32,42)(36,50)(43,48)(46,61)(47,63)(51,52)(54,57)(55,62)(56,60)(59,64);(1,3,8,20)(2,9)(4,22)(5,16,30,17)(6,31,32,50)(7,18,29,19)(10,34,41,45)(11,35)(12,14)(13,38)(15,27)(21,52,40,43)(23,47)(24,56)(25,51)(26,48)(28,44)(33,37)(36,58)(39,60,62,64)(42,49)(46,55)(53,61,57,63)(54,59);(1,4)(2,10)(3,15)(5,23)(6,24)(7,25)(8,27)(9,34)(11,39)(12,40)(13,41)(14,43)(16,46)(17,47)(18,48)(19,51)(20,22)(21,33)(26,29)(28,57)(30,55)(31,59)(32,54)(35,60)(36,61)(37,52)(38,45)(42,62)(44,63)(49,64)(50,56)(53,58);(1,5,29,32,8,30,7,6)(2,11,33,58,13,42,12,28)(3,16,19,50,20,17,18,31)(4,23,26,54,27,55,25,24)(9,35,37,36,38,49,14,44)(10,39,21,53,41,62,40,57)(15,46,51,56,22,47,48,59)(34,60,52,61,45,64,43,63);(1,6,7,30,8,32,29,5)(2,11,33,58,13,42,12,28)(3,17,19,31,20,16,18,50)(4,24,25,55,27,54,26,23)(9,36,14,35,38,44,37,49)(10,39,21,53,41,62,40,57)(15,47,51,59,22,46,48,56)(34,61,43,60,45,63,52,64)Polycyclic presentation:
2^9|b^2=fgh,d^2=fi,e^2=fh,f^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,f^a=fi,g^a=gi,h^a=hi,c^b=ch,e^b=efh,g^b=gi,h^b=hi,g^c=gi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6711045);