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 | C22×D4 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2395 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1678 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)(21,22,26,31,27,30,41,25)(23,32,35,44,34,47,42,28)(24,29,39,37,38,46,52,40)(33,48,49,45,36,50,51,43);(5,7)(6,10)(8,15)(9,14)(11,19)(12,18)(13,16)(17,20)(21,23,27,34)(22,28,30,44)(24,36,38,33)(25,32,31,47)(26,42,41,35)(29,45,46,43)(37,48,40,50)(39,49,52,51);(2,4)(5,8,9,16)(6,11,12,20)(7,13,14,15)(10,17,18,19)(21,24,27,38)(22,29,30,46)(23,33,34,36)(25,40,31,37)(26,39,41,52)(28,43,44,45)(32,48,47,50)(35,49,42,51);(8,16)(11,20)(13,15)(17,19)(21,25,41,30,27,31,26,22)(23,28,42,47,34,44,35,32)(24,37,52,29,38,40,39,46)(33,45,51,48,36,43,49,50);(6,12)(7,14)(11,20)(13,15)(22,30)(23,34)(24,38)(25,31)(35,42)(39,52)(43,45)(48,50)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=ei,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7011865);