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 | #45196 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1524 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)(4,5)(6,8)(9,10,12,14)(11,15,16,13)(17,18)(19,30)(20,26)(21,27)(22,25)(23,38)(24,29)(28,33)(31,45)(32,40)(34,41)(35,39)(36,44)(37,43)(42,46)(47,48);(2,5)(3,8)(9,11)(10,13)(12,16)(14,15)(17,19,21,32,24,33,23,22)(18,25,38,28,29,40,27,30)(20,31,37,47,36,46,34,35)(26,39,41,42,44,48,43,45);(1,3,4,8)(2,6,5,7)(10,14)(11,16)(17,20,24,36)(18,26,29,44)(19,31,33,46)(21,37,23,34)(22,35,32,47)(25,39,40,48)(27,43,38,41)(28,42,30,45);(2,5)(6,7)(10,14)(11,16)(17,21,24,23)(18,27,29,38)(19,32,33,22)(20,34,36,37)(25,30,40,28)(26,41,44,43)(31,35,46,47)(39,42,48,45);(2,5)(6,7)(10,14)(13,15)(17,22,23,33,24,32,21,19)(18,28,27,25,29,30,38,40)(20,35,34,46,36,47,37,31)(26,42,43,39,44,45,41,48)Polycyclic presentation:
2^9|a^2=h,b^2=fgi,c^2=ghi,d^2=i,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, 7514823);