Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3332717 |
| 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 | C2×#392 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2362 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)(9,10)(11,17)(12,16)(13,14)(15,18)(19,24)(20,22)(21,23)(26,27)(28,35)(29,37)(39,40);(6,8)(10,14)(11,18)(16,20)(19,21)(25,26,30,27)(28,31,35,33)(29,32,37,34)(36,39,38,40);(5,6)(7,8)(9,10)(11,15)(12,20)(13,14)(16,22)(17,18)(19,24)(21,23)(25,27,30,26)(28,33,35,31)(29,32,37,34)(36,39,38,40);(1,2,3,4)(9,11)(10,15)(12,21)(13,18)(14,17)(16,24)(19,22)(20,23)(25,28)(26,31)(27,33)(29,38)(30,35)(32,40)(34,39)(36,37);(9,12,13,22)(10,16,14,20)(11,19,18,21)(15,23,17,24)(25,29,30,37)(26,32,27,34)(28,36,35,38)(31,39,33,40)Polycyclic presentation:
2^9|b^2=g,c^2=g,d^2=fi,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3371105);