Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6806623 |
| 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 | |
1338 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)(9,10,12,14)(11,15,16,13)(17,18)(19,30)(20,33)(21,36)(22,28)(23,29)(24,27)(25,32)(26,35)(31,46)(34,44)(37,48)(38,42)(39,45)(40,43)(41,47);(9,11)(10,13)(12,16)(14,15)(17,19)(18,25)(20,21)(22,38)(23,32)(24,35)(26,36)(27,33)(28,44)(29,30)(31,37)(34,39)(40,47)(41,48)(42,45)(43,46);(2,5)(6,8)(10,14)(11,16)(17,20,21,19,23,35,24,32)(18,26,36,25,29,33,27,30)(22,34,37,31,39,38,40,47)(28,42,48,41,45,44,43,46);(10,14)(13,15)(17,21,23,24)(18,27,29,36)(19,20,32,35)(22,37,39,40)(25,33,30,26)(28,43,45,48)(31,34,47,38)(41,44,46,42);(1,3,4,7)(2,6,5,8)(17,22,24,40,23,39,21,37)(18,28,27,43,29,45,36,48)(19,31,20,34,32,47,35,38)(25,41,26,42,30,46,33,44)Polycyclic presentation:
2^9|a^2=gi,c^2=fh,d^2=i,e^2=ghi,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,d^c=di74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6739702);