Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6884841 |
| 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×#121 | D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#529 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1398 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 56 points:
(1,2,4,6)(3,7,8,5)(9,10)(11,19)(12,13)(14,17)(15,18)(16,22)(20,23)(21,24)(25,26,31,32,30,35,27,36)(28,39,37,49,42,46,40,33)(29,43,45,50,44,34,38,47)(41,54,51,48,55,52,53,56);(1,3)(2,5)(4,8)(6,7)(9,11)(10,16)(12,20)(13,21)(14,23)(15,22)(17,24)(18,19)(25,27,30,31)(26,32,35,36)(28,40,42,37)(29,38,44,45)(33,39,49,46)(34,47,43,50)(41,53,55,51)(48,52,56,54);(3,8)(5,7)(9,12,15,17)(10,13,18,14)(11,20,22,24)(16,21,19,23)(25,28,30,42)(26,33,35,49)(27,37,31,40)(29,41,44,55)(32,46,36,39)(34,48,43,56)(38,51,45,53)(47,54,50,52);(2,6)(5,7)(9,13,15,14)(10,17,18,12)(11,21,22,23)(16,24,19,20)(25,29,30,44)(26,34,35,43)(27,38,31,45)(28,41,42,55)(32,47,36,50)(33,48,49,56)(37,51,40,53)(39,52,46,54);(9,14,15,13)(10,17,18,12)(11,21,22,23)(16,20,19,24)(29,44)(34,43)(38,45)(41,55)(47,50)(48,56)(51,53)(52,54)Polycyclic presentation:
2^9|a^2=ghi,b^2=i,c^2=fhi,d^2=fhi,e^2=fh,g^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,d^c=dfh,g^c=gi,e^d=ei65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6900701);