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 | #45092 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2388 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1813 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)(21,22,26,31,27,30,41,48)(23,32,35,44,34,49,25,28)(24,36,40,52,39,29,45,47)(33,51,38,43,37,50,42,46);(5,7)(6,10)(8,15)(9,14)(11,19)(12,18)(13,16)(17,20)(21,23,27,34)(22,28,30,44)(24,37,39,33)(25,41,35,26)(29,46,36,43)(31,49,48,32)(38,45,42,40)(47,50,52,51);(2,4)(5,8)(6,11)(7,13)(9,16)(10,17)(12,20)(14,15)(18,19)(21,24,27,39)(22,29,30,36)(23,33,34,37)(25,42,35,38)(26,40,41,45)(28,43,44,46)(31,47,48,52)(32,50,49,51);(8,16)(11,20)(13,15)(17,19)(21,25,26,23,27,35,41,34)(22,28,31,32,30,44,48,49)(24,38,40,37,39,42,45,33)(29,46,47,51,36,43,52,50);(6,12)(7,14)(11,20)(13,15)(22,30)(23,34)(24,39)(25,35)(31,48)(40,45)(43,46)(50,51)Polycyclic presentation:
2^9|a^2=fhi,b^2=i,c^2=i,d^2=f,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, 6942091);