Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4311730 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2368 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(2,4)(6,8)(9,10,14,15)(11,16,19,23)(12,20,22,13)(17,18,21,24)(25,26)(27,28)(29,37)(30,33)(31,35)(32,38)(34,39)(36,40);(10,15)(13,20)(16,23)(18,24)(25,27,30,35)(26,28,33,31)(29,34,38,40)(32,36,37,39);(5,6,7,8)(9,11)(10,16)(12,21)(13,18)(14,19)(15,23)(17,22)(20,24)(25,27)(26,31)(28,33)(29,34)(30,35)(32,39)(36,37)(38,40);(1,2)(3,4)(9,12)(10,13)(11,17)(14,22)(15,20)(16,24)(18,23)(19,21)(25,28,30,31)(26,27,33,35)(29,36,38,39)(32,34,37,40);(9,13)(10,12)(11,18)(14,20)(15,22)(16,21)(17,23)(19,24)(25,29)(26,32)(27,34)(28,36)(30,38)(31,39)(33,37)(35,40)Polycyclic presentation:
2^9|a^2=f,b^2=h,c^2=gh,d^2=h,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,e^b=ef,d^c=df80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4222719);