Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6432838 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C24 | C22×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#376 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1390 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,4,5)(3,6,7,8)(9,10)(11,14)(12,15)(13,16)(17,21)(18,22)(19,23)(20,24)(27,35)(28,29)(31,38)(33,34)(36,37)(39,40);(2,5)(6,8)(9,11,13,18)(10,14,16,22)(12,19,20,17)(15,23,24,21)(25,26)(27,31)(28,33)(29,34)(30,32)(35,38)(36,39)(37,40);(9,12,13,20)(10,15,16,24)(11,17,18,19)(14,21,22,23)(25,27)(26,31)(28,35)(29,37)(30,36)(32,39)(33,38)(34,40);(1,3,4,7)(2,6,5,8)(10,16)(14,22)(15,24)(21,23)(26,32)(31,39)(33,34)(38,40);(25,28,30,29)(26,33,32,34)(27,35,36,37)(31,38,39,40)Polycyclic presentation:
2^9|a^2=f,b^2=h,c^2=h,d^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dh,e^a=ei,g^a=gi,c^b=ch,d^b=di,g^c=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6341997);