Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #28182 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2535 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| socle | C2 | C2×#2535 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2283 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)(3,15)(4,21)(5,11)(6,12)(7,13)(8,14)(9,19)(10,24)(16,43)(17,38)(18,33)(20,42)(22,50)(23,49)(25,37)(26,36)(27,56)(28,39)(29,40)(30,41)(31,46)(32,47)(34,45)(35,53)(44,59)(48,62)(51,64)(52,57)(54,61)(55,60)(58,63);(1,3)(2,9)(4,22)(5,27)(6,18)(7,19)(8,20)(10,34)(11,38)(12,15)(13,42)(14,33)(16,25)(17,29)(21,31)(23,52)(24,46)(26,45)(28,48)(30,47)(32,40)(35,59)(36,43)(37,50)(39,56)(41,62)(44,55)(49,58)(51,54)(53,63)(57,61)(60,64);(1,4,6,24,8,26,7,25)(2,10,13,21,14,37,12,36)(3,16,19,45,20,46,18,22)(5,23,28,53,30,55,29,54)(9,31,15,34,33,43,42,50)(11,35,40,49,41,61,39,60)(17,44,47,63,48,52,27,51)(32,57,38,58,56,59,62,64);(1,5,7,29,8,30,6,28)(2,11,13,40,14,41,12,39)(3,17,18,27,20,48,19,47)(4,23,25,54,26,55,24,53)(9,32,15,38,33,56,42,62)(10,35,21,49,37,61,36,60)(16,44,22,51,46,52,45,63)(31,57,34,58,43,59,50,64);(1,6,8,7)(2,12,14,13)(3,18,20,19)(4,24,26,25)(5,28,30,29)(9,15,33,42)(10,36,37,21)(11,39,41,40)(16,22,46,45)(17,27,48,47)(23,53,55,54)(31,34,43,50)(32,38,56,62)(35,60,61,49)(44,51,52,63)(57,58,59,64)Polycyclic presentation:
2^9|c^2=g,d^2=h,e^2=i,f^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,d^b=dh,f^b=fi,g^b=gi,h^b=hi,f^c=fi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6370107);