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 |
| prime radical | #48776 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4427 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)(6,8)(9,10,14,17)(11,12,15,19)(13,16,22,24)(18,20,21,23)(25,26,31,32)(27,35,36,33)(28,39,40,29)(30,37,38,34);(3,7)(6,8)(9,11)(10,12)(13,21)(14,15)(16,23)(17,19)(18,22)(20,24)(26,32)(27,36)(29,39)(34,37);(1,2)(3,6)(4,5)(7,8)(11,15)(12,19)(18,21)(20,23)(25,27,31,36)(26,33,32,35)(28,37,40,34)(29,38,39,30);(9,12,14,19)(10,15,17,11)(13,20,22,23)(16,21,24,18)(25,28)(26,29)(27,37)(30,35)(31,40)(32,39)(33,38)(34,36);(1,3)(2,6)(4,7)(5,8)(9,10)(11,12)(13,16)(14,17)(15,19)(18,20)(21,23)(22,24)(25,29)(26,28)(27,38)(30,36)(31,39)(32,40)(33,37)(34,35);(1,2)(3,6)(4,5)(7,8)(9,13)(10,16)(11,18)(12,20)(14,22)(15,21)(17,24)(19,23)(25,30,31,38)(26,34,32,37)(27,29,36,39)(28,35,40,33)Polycyclic presentation:
2^9|a^2=i,c^2=g,d^2=gi,f^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, 8746489);