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 | |
1278 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)(25,26,30,35)(27,32,38,47)(28,40,43,33)(29,34,39,48)(31,36,42,50)(37,53,55,46)(41,52,45,49)(44,56,54,51);(9,11,13,19)(10,14,16,17)(12,18,21,24)(15,22,20,23)(25,27,30,38)(26,32,35,47)(28,41,43,45)(29,42,39,31)(33,49,40,52)(34,50,48,36)(37,54,55,44)(46,56,53,51);(1,3,5,4)(2,6,7,8)(25,28,42,54,30,43,31,44)(26,33,50,56,35,40,36,51)(27,37,29,45,38,55,39,41)(32,46,34,52,47,53,48,49);(9,12,13,21)(10,15,16,20)(11,18,19,24)(14,22,17,23)(28,43)(33,40)(37,55)(41,45)(44,54)(46,53)(49,52)(51,56);(1,4,5,3)(2,6,7,8)(11,19)(14,17)(18,24)(22,23)(25,29,31,27,30,39,42,38)(26,34,36,32,35,48,50,47)(28,41,44,55,43,45,54,37)(33,49,51,53,40,52,56,46)Polycyclic presentation:
2^9|a^2=fi,b^2=fi,c^2=gh,d^2=f,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=di,e^c=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6744408);