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 | |
4540 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(3,7)(6,8)(9,10)(11,19)(12,23)(13,24)(14,18)(15,16)(17,20)(21,22)(25,26)(27,33)(28,36)(29,35)(30,34)(31,32)(37,39)(38,40);(2,5)(6,8)(10,16)(11,20)(12,24)(18,22)(26,32)(29,33)(34,36)(39,40);(1,2)(3,6)(4,5)(7,8)(9,11,15,20)(10,17,16,19)(12,21,24,14)(13,22,23,18)(25,27,31,35)(26,33,32,29)(28,37,30,38)(34,40,36,39);(9,12)(10,13)(11,21)(14,20)(15,24)(16,23)(17,22)(18,19)(25,28,31,30)(26,34,32,36)(27,37,35,38)(29,39,33,40);(1,2)(3,6)(4,5)(7,8)(9,13)(10,12)(11,22)(14,19)(15,23)(16,24)(17,21)(18,20)(25,29,31,33)(26,35,32,27)(28,39,30,40)(34,38,36,37);(1,3)(2,6)(4,7)(5,8)(9,14,15,21)(10,18,16,22)(11,12,20,24)(13,19,23,17)(25,30,31,28)(26,36,32,34)(27,38,35,37)(29,40,33,39)Polycyclic presentation:
2^9|c^2=g,d^2=gh,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^b=egi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8015519);