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 | #2388 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2388 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1791 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)(21,22,28,25,27,32,26,31)(23,33,43,29,36,45,35,46)(24,30,39,42,41,48,40,38)(34,49,50,44,37,52,51,47);(5,7)(6,10)(8,15)(9,14)(11,19)(12,18)(13,16)(17,20)(21,23,27,36)(22,29,32,46)(24,37,41,34)(25,33,31,45)(26,43,28,35)(30,47,48,44)(38,49,42,52)(39,50,40,51);(2,4)(5,8)(6,11)(7,13)(9,16)(10,17)(12,20)(14,15)(18,19)(21,24)(22,30)(23,34)(25,42)(26,40)(27,41)(28,39)(29,44)(31,38)(32,48)(33,49)(35,51)(36,37)(43,50)(45,52)(46,47);(8,16)(11,20)(13,15)(17,19)(21,25,28,32,27,31,26,22)(23,29,43,45,36,46,35,33)(24,38,39,30,41,42,40,48)(34,47,50,49,37,44,51,52);(6,12)(7,14)(11,20)(13,15)(21,26,27,28)(22,25,32,31)(23,35,36,43)(24,39,41,40)(29,45,46,33)(30,38,48,42)(34,50,37,51)(44,49,47,52)Polycyclic presentation:
2^9|a^2=fh,b^2=i,d^2=fi,e^2=i,f^2=i,b^a=bf,c^a=cg,d^a=di,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, 6942202);