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)(28,39)(29,31)(30,34)(33,42)(35,48)(36,45)(37,44)(38,50)(40,49)(41,47)(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)(28,40)(30,36)(32,46)(33,49)(37,51)(39,53)(42,43)(44,48)(47,50)(52,54);(2,6)(7,8)(25,27,36,51,29,37,30,35)(26,32,45,46,31,44,34,48)(28,41,40,52,42,38,43,54)(33,50,53,56,39,47,49,55);(1,3,5,4)(2,7,6,8)(10,15)(14,17)(16,23)(22,24)(26,31)(27,37)(33,39)(34,45)(35,51)(38,41)(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)(25,28,30,43,29,42,36,40)(26,33,34,49,31,39,45,53)(27,38,35,52,37,41,51,54)(32,47,48,56,44,50,46,55)Polycyclic presentation:
2^9|b^2=fi,c^2=hi,d^2=g,e^2=fghi,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, 6750846);