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 | |
4546 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(3,8)(4,6)(9,10)(11,14)(12,15)(13,16)(17,19)(18,21)(20,22)(23,24)(25,26)(27,34)(28,32)(29,37)(30,31)(33,38)(35,36)(39,40);(1,2)(3,4)(5,7)(6,8)(10,11)(12,19)(16,18)(20,24)(26,31)(27,32)(33,37)(35,39);(1,3)(2,6)(4,7)(5,8)(12,19)(13,21)(15,17)(16,18)(26,31)(28,34)(29,38)(35,39);(1,2,5,7)(3,6,8,4)(9,11,14,10)(12,15,19,17)(13,18,21,16)(20,22,24,23)(25,27)(26,28)(29,35)(30,32)(31,34)(33,36)(37,40)(38,39);(1,2)(3,4)(5,7)(6,8)(9,12)(10,15)(11,17)(13,20)(14,19)(16,22)(18,23)(21,24)(25,28)(26,32)(27,31)(29,36)(30,34)(33,39)(35,37)(38,40);(1,4)(2,3)(5,6)(7,8)(9,13,14,21)(10,16,11,18)(12,20,19,24)(15,22,17,23)(25,29,30,38)(26,33,31,37)(27,35,32,39)(28,36,34,40)Polycyclic presentation:
2^9|d^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,e^c=egi,f^c=fgi,e^d=egi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8947630);