Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6806647 |
| 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 | C24 | C22×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1280 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)(3,8)(4,6)(5,7)(9,10,13,16)(11,17,19,14)(12,20,21,15)(18,22,24,23)(25,26)(27,32)(28,40)(29,45)(30,35)(31,36)(33,43)(34,39)(37,53)(38,48)(41,51)(42,50)(44,56)(46,49)(47,55)(52,54);(9,11,13,19)(10,14,16,17)(12,18,21,24)(15,22,20,23)(25,27,30,38)(26,32,35,48)(28,41,43,46)(29,42,39,31)(33,49,40,51)(34,36,45,50)(37,54,55,44)(47,56,53,52);(1,3,5,4)(2,6,7,8)(25,28,42,54,30,43,31,44)(26,33,50,56,35,40,36,52)(27,37,29,46,38,55,39,41)(32,47,45,51,48,53,34,49);(9,12,13,21)(10,15,16,20)(11,18,19,24)(14,22,17,23)(28,43)(33,40)(37,55)(41,46)(44,54)(47,53)(49,51)(52,56);(1,4,5,3)(2,6,7,8)(11,19)(14,17)(18,24)(22,23)(25,29,31,27,30,39,42,38)(26,34,36,48,35,45,50,32)(28,41,44,55,43,46,54,37)(33,51,52,47,40,49,56,53)Polycyclic presentation:
2^9|a^2=f,b^2=fi,c^2=gh,d^2=f,e^2=ghi,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi,d^c=di,e^c=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6744422);