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 | |
4353 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10)(11,16)(12,18)(13,19)(14,20)(15,17)(21,24)(22,23)(25,26,30,35)(27,36,37,31)(28,33,38,32)(29,39,40,34);(1,2,5,6)(3,4,7,8)(9,11)(10,16)(12,13)(14,17)(15,20)(18,19)(21,22)(23,24)(25,27,30,37)(26,31,35,36)(28,34,38,39)(29,33,40,32);(2,6)(3,7)(10,17)(11,20)(13,21)(18,23)(25,28,30,38)(26,32,35,33)(27,34,37,39)(29,36,40,31);(2,6)(3,7)(9,12)(10,18)(11,21)(13,20)(14,19)(15,22)(16,24)(17,23)(25,27)(26,31)(28,34)(29,32)(30,37)(33,40)(35,36)(38,39);(1,3)(2,4)(5,7)(6,8)(9,13,15,21)(10,19,17,24)(11,12,20,22)(14,23,16,18)(25,28)(26,33)(27,34)(29,36)(30,38)(31,40)(32,35)(37,39);(1,4)(2,3)(5,8)(6,7)(9,14)(10,11)(12,19)(13,23)(15,16)(17,20)(18,21)(22,24)(25,29,30,40)(26,34,35,39)(27,32,37,33)(28,36,38,31)Polycyclic presentation:
2^9|a^2=g,b^2=hi,c^2=g,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,f^b=fgi,e^c=egi,e^d=eghi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10236604);