Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #52308 |
| 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 | #52312 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4555 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2)(3,5)(4,6)(7,8)(9,10,13,16)(11,17,19,14)(12,20,21,15)(18,22,24,23)(25,26,30,32)(27,34,36,31)(28,37,29,33)(35,39,38,40);(3,7)(5,8)(9,11,13,19)(10,14,16,17)(12,18,21,24)(15,22,20,23)(25,27,30,36)(26,31,32,34)(28,38,29,35)(33,40,37,39);(1,3)(2,5)(4,7)(6,8)(12,21)(15,20)(18,24)(22,23)(26,32)(31,34)(33,37)(39,40);(3,7)(5,8)(9,12,13,21)(10,15,16,20)(11,18,19,24)(14,22,17,23)(25,28,30,29)(26,33,32,37)(27,35,36,38)(31,39,34,40);(2,6)(5,8)(10,16)(14,17)(15,20)(22,23);(25,29,30,28)(26,33,32,37)(27,35,36,38)(31,40,34,39)Polycyclic presentation:
2^9|a^2=g,b^2=g,d^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,d^c=dghi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7999664);