Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6806623 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #2535 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2395 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1335 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2)(3,7)(4,8)(5,6)(9,10)(11,15)(12,14)(13,16)(17,18)(19,30)(20,25)(21,29)(22,27)(23,28)(24,32)(26,34)(31,45)(33,40)(35,39)(36,44)(37,43)(38,46)(41,48)(42,47);(9,11)(10,13)(12,16)(14,15)(17,19,23,32)(18,24,28,30)(20,34,33,21)(22,35,37,47)(25,29,40,26)(27,42,43,39)(31,36,46,48)(38,44,45,41);(2,6)(7,8)(10,14)(11,16)(17,20,21,19,23,33,34,32)(18,25,29,30,28,40,26,24)(22,35,36,31,37,47,48,46)(27,39,44,45,43,42,41,38);(1,3,5,4)(2,7,6,8)(10,14)(13,15)(17,21,23,34)(18,26,28,29)(19,20,32,33)(22,36,37,48)(24,25,30,40)(27,41,43,44)(31,35,46,47)(38,39,45,42);(1,4,5,3)(2,8,6,7)(17,22,34,48,23,37,21,36)(18,27,26,41,28,43,29,44)(19,31,20,35,32,46,33,47)(24,38,40,42,30,45,25,39)Polycyclic presentation:
2^9|b^2=i,c^2=fhi,d^2=fg,e^2=ghi,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,d^c=di74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6749118);