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 | #2619 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2298 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1253 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)(3,6)(4,5)(7,8)(9,10)(11,14)(12,16)(13,15)(17,21)(18,20)(19,22)(23,24)(25,26,31,33,30,36,29,35)(27,32,42,44,39,47,38,46)(28,34,43,48,40,49,41,50)(37,45,53,54,51,55,52,56);(1,3,7,4)(2,5,8,6)(9,11)(10,14)(12,18)(13,17)(15,21)(16,20)(19,23)(22,24)(25,27,30,39)(26,32,36,47)(28,37,40,51)(29,38,31,42)(33,44,35,46)(34,45,49,55)(41,52,43,53)(48,54,50,56);(1,4,7,3)(2,5,8,6)(10,15)(11,17)(12,19)(20,24)(26,33)(29,31)(32,44)(34,48)(35,36)(38,42)(41,43)(45,54)(46,47)(49,50)(52,53)(55,56);(1,5,7,6)(2,3,8,4)(9,12,13,19)(10,16,15,22)(11,18,17,23)(14,20,21,24)(25,28,30,40)(26,34,36,49)(27,37,39,51)(29,41,31,43)(32,45,47,55)(33,48,35,50)(38,52,42,53)(44,54,46,56);(1,6,7,5)(2,3,8,4)(11,17)(14,21)(18,23)(20,24)(28,40)(34,49)(37,51)(41,43)(45,55)(48,50)(52,53)(54,56)Polycyclic presentation:
2^9|a^2=ghi,b^2=fi,c^2=f,d^2=fhi,e^2=f,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, 6855423);