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 | #2414 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2395 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1696 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)(21,22,26,32,27,31,42,48)(23,33,36,46,35,50,39,28)(24,37,41,52,40,29,47,45)(25,30,34,49,43,44,38,51);(5,7)(6,10)(8,15)(9,14)(11,19)(12,18)(13,16)(17,20)(21,23,27,35)(22,28,31,46)(24,38,40,34)(25,41,43,47)(26,39,42,36)(29,30,37,44)(32,50,48,33)(45,51,52,49);(2,4)(5,8,9,16)(6,11,12,20)(7,13,14,15)(10,17,18,19)(21,24,27,40)(22,29,31,37)(23,34,35,38)(25,36,43,39)(26,41,42,47)(28,44,46,30)(32,45,48,52)(33,51,50,49);(8,16)(11,20)(13,15)(17,19)(21,25,42,38,27,43,26,34)(22,30,48,51,31,44,32,49)(23,24,39,47,35,40,36,41)(28,45,50,29,46,52,33,37);(6,12)(7,14)(11,20)(13,15)(22,31)(23,35)(24,40)(30,44)(32,48)(36,39)(41,47)(49,51)Polycyclic presentation:
2^9|a^2=fhi,b^2=i,c^2=h,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, 7011568);