Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6907945 |
| 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 | C22×D4 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2290 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1333 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 56 points:
(9,10,13,16)(11,17,19,14)(12,20,21,15)(18,22,24,23)(25,26,29,33)(27,35,37,31)(28,39,40,55)(30,43,42,34)(32,46,47,41)(36,50,51,54)(38,45,53,49)(44,52,48,56);(1,2)(3,7)(4,6)(5,8)(9,11)(10,14)(12,18)(13,19)(15,22)(16,17)(20,23)(21,24)(25,27,29,37)(26,31,33,35)(28,36,40,51)(30,38,42,53)(32,44,47,48)(34,45,43,49)(39,54,55,50)(41,52,46,56);(1,3)(2,5)(4,8)(6,7)(12,21)(15,20)(18,24)(22,23)(25,28,30,41,29,40,42,46)(26,32,34,39,33,47,43,55)(27,36,38,52,37,51,53,56)(31,44,45,54,35,48,49,50);(3,8)(5,7)(9,12,13,21)(10,15,16,20)(11,18,19,24)(14,22,17,23)(26,33)(27,37)(28,40)(34,43)(38,53)(41,46)(44,48)(50,54);(2,6)(5,7)(11,19)(14,17)(18,24)(22,23)(27,37)(28,40)(31,35)(32,47)(38,53)(39,55)(41,46)(45,49)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=g,d^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,d^c=dfh,e^c=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6817073);