Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6433057 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#540 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1414 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2)(3,6)(4,5)(7,8)(9,10)(11,14)(12,16)(13,15)(17,20)(18,21)(19,22)(23,24)(25,26)(27,37)(28,40)(29,39)(30,34)(31,35)(32,36)(33,38);(2,5)(6,8)(9,11)(10,14)(12,18)(13,17)(15,20)(16,21)(19,23)(22,24)(25,27,31,38,32,39,30,28)(26,33,34,37,36,40,35,29);(1,3,4,7)(2,6,5,8)(11,17)(14,20)(18,23)(21,24)(25,28,31,27,32,38,30,39)(26,29,34,33,36,37,35,40);(10,15)(14,20)(16,22)(21,24)(25,29,32,37)(26,27,36,39)(28,34,38,35)(30,40,31,33);(9,12,13,19)(10,16,15,22)(11,18,17,23)(14,21,20,24)(25,30,32,31)(26,34,36,35)(27,28,39,38)(29,33,37,40)Polycyclic presentation:
2^9|b^2=gi,c^2=f,d^2=i,e^2=h,f^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dh,f^a=fi,g^a=gi,c^b=ch,d^b=di,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6344402);