Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6993947 |
| 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 | C2×#121 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1664 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)(8,16)(11,17)(15,20)(18,19)(21,22)(23,33)(24,38)(25,30)(26,31)(27,43)(28,32)(29,36)(34,51)(35,44)(37,46)(39,45)(40,50)(41,49)(42,47)(48,52);(5,6,9,12)(7,13,14,10)(8,17,16,11)(15,18,20,19)(21,23,27,36,28,37,26,42)(22,29,31,33,32,47,43,46)(24,39,40,35,25,34,41,52)(30,48,50,51,38,44,49,45);(2,4)(5,7)(6,10)(8,15)(9,14)(11,18)(12,13)(16,20)(17,19)(21,24)(22,30)(23,34)(25,28)(26,41)(27,40)(29,44)(31,50)(32,38)(33,45)(35,42)(36,52)(37,39)(43,49)(46,51)(47,48);(7,14)(10,13)(15,20)(18,19)(21,25,28,24)(22,30,32,38)(23,35,37,52)(26,41,27,40)(29,45,47,51)(31,49,43,50)(33,44,46,48)(34,36,39,42);(5,8,9,16)(6,11,12,17)(7,15,14,20)(10,18,13,19)(21,26,28,27)(22,31,32,43)(23,36,37,42)(24,40,25,41)(29,46,47,33)(30,49,38,50)(34,35,39,52)(44,45,48,51)Polycyclic presentation:
2^9|a^2=gi,b^2=fhi,d^2=i,e^2=h,f^2=i,b^a=bf,c^a=cg,e^a=ehi,f^a=fi,c^b=ch,d^b=df,e^b=eh,d^c=dh,e^c=ei,f^d=fi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7089967);