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 | |
1162 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)(25,26,29,33)(27,35,37,31)(28,32,41,47)(30,34,42,48)(36,49,50,43)(38,46,40,44)(39,45,52,55)(51,56,53,54);(9,11,13,19)(10,14,15,17)(12,20,21,18)(16,24,23,22)(27,36)(28,39)(30,42)(31,43)(32,45)(34,48)(35,49)(37,50)(38,40)(41,52)(44,46)(47,55);(2,6)(7,8)(25,27,30,36,29,37,42,50)(26,31,34,43,33,35,48,49)(28,40,39,53,41,38,52,51)(32,46,45,56,47,44,55,54);(1,3,5,4)(2,7,6,8)(10,15)(14,17)(16,23)(22,24)(27,37)(31,35)(36,50)(38,40)(43,49)(44,46)(51,53)(54,56);(1,4,5,3)(2,8,6,7)(9,12,13,21)(10,16,15,23)(11,18,19,20)(14,22,17,24)(25,28,30,39,29,41,42,52)(26,32,34,45,33,47,48,55)(27,38,36,51,37,40,50,53)(31,44,43,54,35,46,49,56)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=h,d^2=gi,e^2=fghi,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, 6750724);