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, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2096 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 40 points:
(2,4)(9,10)(11,12)(13,16)(14,15)(17,20)(18,19)(21,22)(23,24)(25,26,30,33)(27,28,35,31)(29,37,39,32)(34,40,38,36);(6,8)(10,15)(11,17)(13,21)(19,24)(25,27,30,35)(26,31,33,28)(29,38,39,34)(32,36,37,40);(5,6,7,8)(9,11,14,17)(10,12,15,20)(13,18,21,23)(16,19,22,24)(25,26)(27,31)(28,35)(29,32)(30,33)(34,40)(36,38)(37,39);(1,2)(3,4)(9,12)(10,11)(13,19)(14,20)(15,17)(16,18)(21,24)(22,23)(25,28,30,31)(26,27,33,35)(29,36,39,40)(32,34,37,38);(9,13,14,21)(10,16,15,22)(11,18,17,23)(12,19,20,24)(25,29,30,39)(26,32,33,37)(27,34,35,38)(28,36,31,40)Polycyclic presentation:
2^9|a^2=g,b^2=g,c^2=h,d^2=g,e^2=f,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, 1479981);