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 | |
4499 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,4,6)(3,5,7,8)(9,10,14,15)(11,18,19,16)(12,13,17,23)(20,21,22,24);(1,3,4,7)(2,5,6,8)(10,15)(11,19)(12,17)(21,24)(25,26)(27,33)(28,36)(29,30)(31,35)(32,37)(34,39)(38,40);(1,2)(3,5)(4,6)(7,8)(12,17)(13,23)(20,22)(21,24)(26,30)(28,37)(31,33)(34,38);(9,11,14,19)(10,16,15,18)(12,22,17,20)(13,21,23,24)(25,27)(26,31)(28,38)(29,35)(30,33)(32,40)(34,37)(36,39);(1,3)(2,5)(4,7)(6,8)(9,12,14,17)(10,13,15,23)(11,20,19,22)(16,24,18,21)(25,28)(26,32)(27,34)(29,37)(30,36)(31,39)(33,40)(35,38);(1,2)(3,5)(4,6)(7,8)(9,13)(10,17)(11,21)(12,15)(14,23)(16,20)(18,22)(19,24)(25,26)(27,31)(28,32)(29,30)(33,35)(34,39)(36,37)(38,40)Polycyclic presentation:
2^9|a^2=gh,b^2=h,d^2=g,e^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, 8947421);