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 | |
4391 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,6,5)(3,8,4,7)(9,10,15,19)(11,20,21,16)(12,17,23,22)(13,14,18,24)(25,26)(27,33)(28,35)(29,32)(30,31)(34,37)(36,40)(38,39);(1,3,6,4)(2,7,5,8)(11,21)(12,23)(16,20)(17,22)(26,31)(28,33)(32,39)(36,37);(9,11,15,21)(10,16,19,20)(12,18,23,13)(14,22,24,17)(25,27)(26,28)(29,34)(30,35)(31,33)(32,36)(37,39)(38,40);(2,5)(7,8)(9,12)(10,17)(11,18)(13,21)(14,16)(15,23)(19,22)(20,24)(25,28)(26,27)(29,37)(30,33)(31,35)(32,40)(34,39)(36,38);(1,4,6,3)(2,8,5,7)(9,13,15,18)(10,14,19,24)(11,12,21,23)(16,22,20,17);(1,5,6,2)(3,8,4,7)(9,14)(10,18)(11,22)(12,20)(13,19)(15,24)(16,23)(17,21)(25,29,30,38)(26,32,31,39)(27,34,35,40)(28,36,33,37)Polycyclic presentation:
2^9|a^2=gh,b^2=ghi,c^2=i,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8344029);