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 | |
4714 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)(3,7)(9,10)(11,17)(12,20)(13,14)(15,18)(16,22)(19,23)(21,24)(25,26)(27,28)(29,37)(30,31)(32,33)(34,38)(35,39)(36,40);(2,5)(3,7)(10,14)(15,17)(16,20)(23,24)(26,31)(28,32)(29,38)(35,40);(1,2)(3,6)(4,5)(7,8)(11,18)(15,17)(19,21)(23,24)(25,27,30,33)(26,32,31,28)(29,35,38,40)(34,39,37,36);(2,5)(6,8)(9,11)(10,15)(12,21)(13,18)(14,17)(16,24)(19,22)(20,23)(25,27)(26,32)(28,31)(29,35)(30,33)(34,39)(36,37)(38,40);(1,2)(3,6)(4,5)(7,8)(9,12,13,22)(10,16,14,20)(11,19,18,21)(15,23,17,24)(25,28,30,32)(26,33,31,27)(29,36,38,39)(34,40,37,35);(1,3)(2,6)(4,7)(5,8)(9,10)(11,15)(12,16)(13,14)(17,18)(19,23)(20,22)(21,24)(25,29)(26,34)(27,35)(28,36)(30,38)(31,37)(32,39)(33,40)Polycyclic presentation:
2^9|c^2=hi,e^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^d=egi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8736349);