Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7518920 |
| 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 | #25388 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2395 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1396 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)(4,6)(5,8)(9,10,13,12)(11,14,16,15)(17,18)(19,29)(20,25)(21,24)(22,37)(23,28)(26,32)(27,30)(31,43)(33,40)(34,38)(35,47)(36,39)(41,45)(42,44)(46,48);(1,3,4,8)(2,5,6,7)(9,11)(10,14)(12,15)(13,16)(17,19,30,36,23,32,22,21)(18,24,37,26,28,39,27,29)(20,31,44,46,33,45,35,34)(25,38,47,41,40,48,42,43);(2,6)(3,8)(9,12,13,10)(11,15,16,14)(17,20,23,33)(18,25,28,40)(19,31,32,45)(21,34,36,46)(22,35,30,44)(24,38,39,48)(26,41,29,43)(27,42,37,47);(2,6)(5,7)(11,16)(14,15)(20,33)(25,40)(31,45)(34,46)(35,44)(38,48)(41,43)(42,47);(2,6)(5,7)(17,21,22,32,23,36,30,19)(18,26,27,24,28,29,37,39)(20,34,35,45,33,46,44,31)(25,41,42,38,40,43,47,48)Polycyclic presentation:
2^9|a^2=h,b^2=fi,c^2=hi,e^2=fg,f^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,f^a=fi,c^b=cg,d^b=dh,d^c=di80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7514117);