Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1506685 |
| automorphism | ? |
| 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 | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2198 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)(12,19)(13,21)(14,15)(16,22)(18,20)(23,24)(26,28)(31,35)(32,36)(38,39);(6,8)(10,15)(12,17)(13,22)(18,24)(25,26,30,28)(27,31,34,35)(29,36,37,32)(33,39,40,38);(5,6,7,8)(9,11,14,19)(10,12,15,17)(13,18,22,24)(16,20,21,23)(25,27)(26,31)(28,35)(29,33)(30,34)(32,38)(36,39)(37,40);(1,2,3,4)(9,12)(10,11)(13,20)(14,17)(15,19)(16,18)(21,24)(22,23)(25,28,30,26)(27,31,34,35)(29,36,37,32)(33,38,40,39);(9,13)(10,16)(11,18)(12,20)(14,22)(15,21)(17,23)(19,24)(25,29,30,37)(26,32,28,36)(27,33,34,40)(31,38,35,39)Polycyclic presentation:
2^9|b^2=fg,c^2=gh,d^2=gi,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^b=ef,d^c=df74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1422526);