Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6792404 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #38904 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1794 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 64 points:
(1,2,8,7)(3,9,16,27)(4,10,20,28)(5,11,22,23)(6,12,26,25)(13,17,37,29)(14,30,39,49)(15,24,34,51)(18,42,44,31)(19,32,45,52)(21,33,47,48)(35,53,58,40)(36,41,56,50)(38,46,55,63)(43,60,61,54)(57,62,64,59);(1,3,8,16)(2,9,7,27)(4,17,20,29)(5,14,22,39)(6,24,26,51)(10,13,28,37)(11,30,23,49)(12,34,25,15)(18,40,44,53)(19,36,45,56)(21,46,47,63)(31,35,42,58)(32,50,52,41)(33,55,48,38)(43,57,61,64)(54,62,60,59);(1,4)(2,10)(3,13)(5,18)(6,19)(7,28)(8,20)(9,29)(11,31)(12,32)(14,35)(15,36)(16,37)(17,27)(21,43)(22,44)(23,42)(24,50)(25,52)(26,45)(30,53)(33,54)(34,56)(38,57)(39,58)(40,49)(41,51)(46,62)(47,61)(48,60)(55,64)(59,63);(1,5)(2,11)(3,14)(4,18)(6,21)(7,23)(8,22)(9,30)(10,31)(12,33)(13,35)(15,38)(16,39)(17,40)(19,43)(20,44)(24,46)(25,48)(26,47)(27,49)(28,42)(29,53)(32,54)(34,55)(36,57)(37,58)(41,59)(45,61)(50,62)(51,63)(52,60)(56,64);(1,6,2,12,8,26,7,25)(3,15,27,51,16,34,9,24)(4,19,10,32,20,45,28,52)(5,21,11,33,22,47,23,48)(13,36,17,41,37,56,29,50)(14,38,49,63,39,55,30,46)(18,43,31,54,44,61,42,60)(35,57,40,59,58,64,53,62)Polycyclic presentation:
2^9|a^2=i,b^2=i,e^2=fhi,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,g^b=gi,h^b=hi,f^c=fi,h^c=hi,g^d=gi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6699605);