Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667556 |
| 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×#449 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1459 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)(9,10)(11,14)(12,13)(15,16)(17,18,23,28)(19,24,31,39)(20,25,22,27)(21,26,34,42)(29,37,35,43)(30,38,46,48)(32,40,36,44)(33,41,45,47);(1,3)(2,5)(4,8)(6,7)(17,19,23,31)(18,24,28,39)(20,29,22,35)(21,33,34,45)(25,37,27,43)(26,41,42,47)(30,36,46,32)(38,44,48,40);(2,6)(5,7)(10,13)(14,16)(17,20)(18,25)(21,32)(22,23)(26,40)(27,28)(29,35)(33,45)(34,36)(37,43)(41,47)(42,44);(2,6)(5,7)(9,11,12,15)(10,14,13,16)(17,20,23,22)(18,25,28,27)(19,29,31,35)(21,32,34,36)(24,37,39,43)(26,40,42,44)(30,45,46,33)(38,47,48,41);(3,8)(5,7)(17,21,23,34)(18,26,28,42)(19,30,31,46)(20,32,22,36)(24,38,39,48)(25,40,27,44)(29,45,35,33)(37,47,43,41)Polycyclic presentation:
2^9|a^2=fi,b^2=i,d^2=fgi,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,d^c=di,h^c=hi,h^e=hi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6606843);