Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4556 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)(3,7)(4,6)(5,8)(9,10)(11,15)(12,16)(13,17)(14,19)(18,22)(20,23)(21,24)(25,26,30,32)(27,31,33,28)(29,34,38,39)(35,37,40,36);(1,2,4,6)(3,5,8,7)(9,11,13,19)(10,14,17,15)(12,18,20,24)(16,21,23,22)(25,27,30,33)(26,31,32,28)(29,35,38,40)(34,37,39,36);(1,2)(3,5)(4,6)(7,8)(12,20)(16,23)(18,24)(21,22)(26,32)(27,33)(34,39)(35,40);(1,3)(2,5)(4,8)(6,7)(9,10)(11,14)(12,16)(13,17)(15,19)(18,21)(20,23)(22,24)(25,28,30,31)(26,33,32,27)(29,37,38,36)(34,35,39,40);(9,11)(10,15)(12,18)(13,19)(14,17)(16,22)(20,24)(21,23)(25,29,30,38)(26,34,32,39)(27,35,33,40)(28,36,31,37);(9,12)(10,16)(11,18)(13,20)(14,21)(15,22)(17,23)(19,24)(25,26)(27,31)(28,33)(29,34)(30,32)(35,37)(36,40)(38,39)Polycyclic presentation:
2^9|a^2=h,b^2=gh,d^2=h,e^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,f^c=fgi,e^d=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8832981);