Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #140091 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1918 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| socle | C2 | #11379 |
| prime radical | C24 | #135 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
435 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 64 points:
(1,2,8,15,35,49,6,13,10,17,7,14,9,16,5,33)(3,18,23,11,36,40,20,50,25,47,22,42,24,39,21,41)(4,26,30,46,37,61,29,45,32,48,34,12,31,44,28,51)(19,53,27,52,59,60,54,62,57,38,58,43,56,64,55,63);(1,3)(2,11,17,42)(4,27)(5,23)(6,22)(7,20)(8,21)(9,36)(10,25)(12,43,46,52)(13,39,33,40)(14,47,15,18)(16,50,49,41)(19,30)(24,35)(26,60,48,64)(28,59)(29,56)(31,54)(32,58)(34,57)(37,55)(38,61,53,44)(45,63,51,62);(1,4)(2,12,17,46)(3,19)(5,34)(6,30)(7,28)(8,29)(9,37)(10,32)(11,38,42,53)(13,44,33,61)(14,48,15,26)(16,51,49,45)(18,52,47,43)(20,27)(21,58)(22,55)(23,54)(24,59)(25,57)(31,35)(36,56)(39,62,40,63)(41,64,50,60)Polycyclic presentation:
2^9|a^2=g,b^2=deh,c^2=deh,d^2=h,e^2=h,f^2=hi,g^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=di,c^b=cf,d^b=dh,e^b=eh,f^b=fh,g^b=gh,h^b=hi,d^c=dh,e^c=eh,f^c=fh,g^c=ghi,h^c=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 56607);