Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1506685 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26 |
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 | |
1676 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,4,6)(3,7,8,5)(11,15)(12,16)(17,18,21,20)(19,22,23,24)(25,26,30,31)(27,32,33,28)(29,34,37,38)(35,39,40,36);(3,8)(5,7)(9,10,13,14)(11,16,15,12)(19,23)(22,24)(26,31)(27,33)(29,37)(36,39);(1,2)(3,5)(4,6)(7,8)(9,11)(10,12)(13,15)(14,16)(17,19)(18,22)(20,24)(21,23)(25,27)(26,32)(28,31)(29,35)(30,33)(34,39)(36,38)(37,40);(1,3)(2,5)(4,8)(6,7)(9,10)(11,12)(13,14)(15,16)(17,20,21,18)(19,24,23,22)(25,28,30,32)(26,33,31,27)(29,36,37,39)(34,40,38,35);(1,2)(3,5)(4,6)(7,8)(9,12)(10,11)(13,16)(14,15)(25,29,30,37)(26,34,31,38)(27,35,33,40)(28,36,32,39)Polycyclic presentation:
2^9|a^2=gh,b^2=fgi,d^2=fghi,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, 1622214);