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 ◁ #2401 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1166 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)(3,7)(4,8)(5,6)(9,10)(11,17)(12,16)(13,15)(14,19)(18,24)(20,22)(21,23)(25,26)(27,32)(28,38)(29,31)(30,34)(33,42)(35,48)(36,44)(37,50)(39,49)(40,47)(41,45)(43,53)(46,51)(52,55)(54,56);(9,11,13,19)(10,14,15,17)(12,20,21,18)(16,24,23,22)(26,31)(27,35)(28,39)(30,41)(32,46)(33,49)(36,51)(38,53)(42,43)(44,48)(47,50)(52,54);(2,6)(7,8)(25,27,30,35,29,36,41,51)(26,32,34,48,31,44,45,46)(28,40,43,54,42,37,39,52)(33,50,49,55,38,47,53,56);(1,3,5,4)(2,7,6,8)(10,15)(14,17)(16,23)(22,24)(26,31)(27,36)(33,38)(34,45)(35,51)(37,40)(49,53)(52,54);(1,4,5,3)(2,8,6,7)(9,12,13,21)(10,16,15,23)(11,18,19,20)(14,22,17,24)(25,28,41,39,29,42,30,43)(26,33,45,53,31,38,34,49)(27,37,51,54,36,40,35,52)(32,47,46,55,44,50,48,56)Polycyclic presentation:
2^9|b^2=fi,c^2=h,d^2=g,e^2=fgh,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, 6750841);