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×D8 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2399 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1694 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)(21,22,27,33)(23,35,39,29)(24,41,25,31)(26,34,28,32)(30,43,48,42)(36,45,37,50)(38,46,40,47)(44,51,49,52);(2,3)(5,6,9,12)(7,13,14,10)(8,17,16,11)(15,18,20,19)(21,23,27,39)(22,29,33,35)(24,36,25,37)(26,38,28,40)(30,44,48,49)(31,45,41,50)(32,46,34,47)(42,51,43,52);(5,7)(6,10)(8,15)(9,14)(11,18)(12,13)(16,20)(17,19)(21,24,27,25)(22,30,33,48)(23,36,39,37)(26,42,28,43)(29,44,35,49)(31,32,41,34)(38,51,40,52)(45,46,50,47);(1,3,4,2)(7,14)(10,13)(15,20)(18,19)(21,25,27,24)(22,31,33,41)(23,37,39,36)(26,42,28,43)(29,45,35,50)(30,32,48,34)(38,51,40,52)(44,46,49,47);(5,8,9,16)(6,11,12,17)(7,15,14,20)(10,18,13,19)(21,26,27,28)(22,32,33,34)(23,38,39,40)(24,42,25,43)(29,46,35,47)(30,41,48,31)(36,51,37,52)(44,50,49,45)Polycyclic presentation:
2^9|a^2=f,b^2=hi,c^2=i,d^2=f,e^2=hi,g^2=i,b^a=bf,c^a=cg,e^a=ehi,g^a=gi,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, 7088350);