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 | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2259 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)(9,10,14,15)(11,16,20,23)(12,21,22,17)(13,18,24,19)(26,27)(28,35)(29,37)(39,40);(6,8)(10,15)(12,22)(16,23)(18,19)(25,26,30,27)(28,31,35,33)(29,32,37,34)(36,39,38,40);(5,6)(7,8)(9,11)(10,16)(12,19)(13,21)(14,20)(15,23)(17,24)(18,22)(25,27,30,26)(28,33,35,31)(29,32,37,34)(36,39,38,40);(1,2,3,4)(9,12,14,22)(10,17,15,21)(11,19,20,18)(13,16,24,23)(25,28)(26,31)(27,33)(29,38)(30,35)(32,40)(34,39)(36,37);(9,13,14,24)(10,18,15,19)(11,17,20,21)(12,23,22,16)(25,29,30,37)(26,32,27,34)(28,36,35,38)(31,39,33,40)Polycyclic presentation:
2^9|a^2=fg,b^2=g,c^2=g,d^2=gi,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, 3371250);