Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996639 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #45236 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2388 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1806 conjugacy classes of subgroups.
| minimal degree | μ(G) = 52 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 52 points:
(1,2)(3,4)(5,6,9,12)(7,10,14,18)(8,11,16,20)(13,17,15,19)(21,22,26,31,27,32,40,48)(23,33,35,45,36,50,37,28)(24,29,39,47,38,46,52,44)(25,30,42,49,41,43,34,51);(5,7)(6,10)(8,15)(9,14)(11,19)(12,18)(13,16)(17,20)(21,23,27,36)(22,28,32,45)(24,34,38,42)(25,39,41,52)(26,37,40,35)(29,43,46,30)(31,50,48,33)(44,51,47,49);(2,4)(5,8)(6,11)(7,13)(9,16)(10,17)(12,20)(14,15)(18,19)(21,24,27,38)(22,29,32,46)(23,34,36,42)(25,37,41,35)(26,39,40,52)(28,43,45,30)(31,47,48,44)(33,51,50,49);(8,16)(11,20)(13,15)(17,19)(21,25,40,34,27,41,26,42)(22,30,48,51,32,43,31,49)(23,24,37,52,36,38,35,39)(28,44,50,46,45,47,33,29);(6,12)(7,14)(11,20)(13,15)(24,38)(25,41)(29,46)(30,43)(34,42)(39,52)(44,47)(49,51)Polycyclic presentation:
2^9|a^2=fh,b^2=i,c^2=i,d^2=fi,f^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,e^c=ei,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6942077);