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 | C22×D4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2399 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1647 conjugacy classes of subgroups.
| minimal degree | μ(G) = 52 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 52 points:
(1,2,4,3)(8,16)(11,17)(15,20)(18,19)(21,22,27,41,26,32,25,31)(23,28,34,44,36,46,35,45)(24,30,38,29,40,48,39,49)(33,43,52,42,37,50,51,47);(2,3)(5,6,9,12)(7,13,14,10)(8,17,16,11)(15,18,20,19)(21,23,26,36)(22,28,32,46)(24,37,40,33)(25,35,27,34)(29,47,49,42)(30,50,48,43)(31,45,41,44)(38,51,39,52);(5,7)(6,10)(8,15)(9,14)(11,18)(12,13)(16,20)(17,19)(21,24,26,40)(22,29,32,49)(23,33,36,37)(25,38,27,39)(28,42,46,47)(30,31,48,41)(34,51,35,52)(43,45,50,44);(1,3,4,2)(7,14)(10,13)(15,20)(18,19)(21,24)(22,30)(23,33)(25,39)(26,40)(27,38)(28,43)(29,41)(31,49)(32,48)(34,52)(35,51)(36,37)(42,44)(45,47)(46,50);(5,8,9,16)(6,11,12,17)(7,15,14,20)(10,18,13,19)(21,25,26,27)(22,31,32,41)(23,34,36,35)(24,38,40,39)(28,44,46,45)(29,48,49,30)(33,51,37,52)(42,43,47,50)Polycyclic presentation:
2^9|a^2=fgi,b^2=h,c^2=i,d^2=f,e^2=h,g^2=i,b^a=bf,c^a=cg,e^a=ehi,c^b=ch,d^b=dfi,e^b=eh,d^c=dh,g^c=gi,e^d=ei65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7091057);