Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996355 |
| 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 | #45412 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1778 conjugacy classes of subgroups.
| minimal degree | μ(G) = 52 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 52 points:
(2,4)(5,6,9,12)(7,10,14,18)(8,11,16,20)(13,17,15,19)(21,22,25,31,26,30,37,41)(23,28,34,43,33,45,40,27)(24,29,38,48,39,47,49,52)(32,44,35,50,36,46,51,42);(5,7)(6,10)(8,15)(9,14)(11,19)(12,18)(13,16)(17,20)(21,23,26,33)(22,27,30,43)(24,35,39,51)(25,40,37,34)(28,31,45,41)(29,44,47,46)(32,49,36,38)(42,52,50,48);(1,2,3,4)(5,8)(6,11)(7,13)(9,16)(10,17)(12,20)(14,15)(18,19)(21,23)(22,28)(24,36)(25,34)(26,33)(27,41)(29,46)(30,45)(31,43)(32,39)(35,49)(37,40)(38,51)(42,48)(44,47)(50,52);(8,16)(11,20)(13,15)(17,19)(21,24,37,49,26,39,25,38)(22,29,41,52,30,47,31,48)(23,32,40,51,33,36,34,35)(27,42,45,46,43,50,28,44);(6,12)(7,14)(11,20)(13,15)(22,30)(23,33)(29,47)(31,41)(32,36)(34,40)(35,51)(48,52)Polycyclic presentation:
2^9|a^2=fhi,b^2=i,c^2=g,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=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6949393);