Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6438991 |
| 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 | C2×#469 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2298 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1417 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,4,5)(3,6,7,8)(9,10)(11,14)(12,13)(15,16)(17,18,22,28,24,30,23,29)(19,31,20,25,34,42,33,26)(21,27,37,44,39,46,38,45)(32,43,35,40,47,48,36,41);(2,5)(6,8)(17,19,24,34)(18,25,30,26)(20,22,33,23)(21,36,39,35)(27,43,46,48)(28,31,29,42)(32,37,47,38)(40,44,41,45);(1,3,4,7)(2,6,5,8)(10,13)(14,16)(17,20,23,19,24,33,22,34)(18,26,29,42,30,25,28,31)(21,36,38,47,39,35,37,32)(27,40,45,43,46,41,44,48);(17,21,22,37,24,39,23,38)(18,27,28,44,30,46,29,45)(19,32,20,35,34,47,33,36)(25,40,42,48,26,41,31,43);(9,11,12,15)(10,14,13,16)(17,22,24,23)(18,28,30,29)(19,33,34,20)(21,37,39,38)(25,31,26,42)(27,44,46,45)(32,36,47,35)(40,43,41,48)Polycyclic presentation:
2^9|a^2=fi,b^2=i,c^2=f,d^2=hi,e^2=g,f^2=i,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,f^b=fi,h^b=hi,d^c=di,e^c=ei92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6372126);