Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996375 |
| 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 | #2431 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1470 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,4,6)(3,7,8,5)(10,11)(12,16)(17,18,22,26,23,27,35,43)(19,28,30,38,31,42,21,24)(20,25,34,41,33,40,46,47)(29,44,32,39,45,48,36,37);(1,3)(2,5)(4,8)(6,7)(9,10)(11,13)(12,15)(14,16)(17,19,23,31)(18,24,27,38)(20,29,33,45)(21,35,30,22)(25,37,40,39)(26,42,43,28)(32,34,36,46)(41,48,47,44);(2,6)(5,7)(9,11,13,10)(12,15,16,14)(17,20,23,33)(18,25,27,40)(19,29,31,45)(21,36,30,32)(22,34,35,46)(24,37,38,39)(26,41,43,47)(28,44,42,48);(3,8)(5,7)(10,11)(14,15)(17,21,22,19,23,30,35,31)(18,24,26,28,27,38,43,42)(20,32,34,45,33,36,46,29)(25,39,41,48,40,37,47,44);(9,12,13,16)(10,14,11,15)(20,33)(25,40)(29,45)(32,36)(34,46)(37,39)(41,47)(44,48)Polycyclic presentation:
2^9|a^2=fh,b^2=i,c^2=hi,d^2=fg,e^2=h,f^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dhi,e^c=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7104737);