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 | |
4506 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,5)(6,7)(9,10)(11,16)(12,14)(13,20)(15,19)(17,21)(18,22)(23,24)(25,26)(27,28)(29,32)(30,33)(31,35)(34,36)(37,38)(39,40);(1,2,4,5)(3,6,8,7)(9,11)(10,15)(12,16)(13,18)(14,19)(17,22)(20,23)(21,24)(27,31)(28,35)(34,39)(36,40);(9,10,14,12)(11,16,19,15)(13,17,21,20)(18,23,24,22)(25,27,30,31)(26,28,33,35)(29,34,38,39)(32,36,37,40);(1,2,4,5)(3,7,8,6)(9,12,14,10)(11,16,19,15)(13,20,21,17)(18,23,24,22)(25,26)(27,28)(29,37)(30,33)(31,35)(32,38)(34,40)(36,39);(1,2)(3,6)(4,5)(7,8)(9,11)(10,16)(12,15)(13,18)(14,19)(17,23)(20,22)(21,24)(25,28)(26,31)(27,33)(29,36)(30,35)(32,39)(34,37)(38,40);(1,3,4,8)(2,6,5,7)(9,13,14,21)(10,17,12,20)(11,18,19,24)(15,22,16,23)(25,29)(26,32)(27,34)(28,36)(30,38)(31,39)(33,37)(35,40)Polycyclic presentation:
2^9|b^2=gh,c^2=i,d^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=egi,e^d=eghi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10330207);