Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #45438 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C25 |
| commutator, Frattini, socle, cosocle | C23 | C26 |
| prime radical | #45481 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4380 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2)(3,6)(4,5)(7,8)(9,10)(11,17)(12,15)(13,16)(14,19)(18,23)(20,22)(21,24)(25,26)(27,30)(28,35)(29,32)(31,37)(33,40)(34,39)(36,38);(2,5)(6,8)(25,27,29,34)(26,30,32,39)(28,36,37,33)(31,40,35,38);(9,11,13,19)(10,14,16,17)(12,20,21,18)(15,23,24,22)(25,28,29,37)(26,31,32,35)(27,33,34,36)(30,38,39,40);(9,12,13,21)(10,15,16,24)(11,18,19,20)(14,22,17,23)(26,32)(30,39)(31,35)(38,40);(10,16)(14,17)(15,24)(22,23)(26,32)(28,37)(30,39)(33,36);(1,3,4,7)(2,6,5,8)Polycyclic presentation:
2^9|b^2=i,c^2=h,d^2=hi,f^2=g,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^c=dhi,e^c=ei104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7610086);