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 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2399 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1714 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)(7,10,13,17)(8,15,16,11)(14,20,19,18)(21,22,26,31,27,32,43,50)(23,33,36,46,37,49,39,28)(24,38,40,45,35,29,41,47)(25,30,34,48,42,44,52,51);(7,13)(8,16)(10,17)(11,15)(21,23,27,37)(22,28,32,46)(24,34,35,52)(25,41,42,40)(26,39,43,36)(29,44,38,30)(31,49,50,33)(45,51,47,48);(1,2,3,4)(5,7,9,13)(6,10,12,17)(8,14,16,19)(11,18,15,20)(21,24)(22,29)(23,34)(25,39)(26,40)(27,35)(28,44)(30,46)(31,47)(32,38)(33,51)(36,42)(37,52)(41,43)(45,50)(48,49);(7,13)(10,17)(14,19)(18,20)(21,25,26,34,27,42,43,52)(22,30,31,48,32,44,50,51)(23,35,36,41,37,24,39,40)(28,45,33,29,46,47,49,38);(5,8,9,16)(6,11,12,15)(7,14,13,19)(10,18,17,20)(24,35)(25,42)(29,38)(30,44)(34,52)(40,41)(45,47)(48,51)Polycyclic presentation:
2^9|a^2=fh,b^2=i,c^2=ghi,d^2=f,e^2=h,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, 7103991);