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 | #56855 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4175 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,6,7)(3,8,4,5)(9,10,14,12)(11,15,17,18)(13,16,22,20)(19,23,21,24)(26,28)(31,33)(32,36)(38,39);(2,7)(3,4)(9,11)(10,15)(12,18)(13,21)(14,17)(16,24)(19,22)(20,23)(25,26,30,28)(27,33,35,31)(29,32,37,36)(34,39,40,38);(2,7)(5,8)(11,17)(15,18)(19,21)(23,24)(25,27,30,35)(26,31,28,33)(29,34,37,40)(32,38,36,39);(1,3,6,4)(2,5,7,8)(9,12,14,10)(11,18,17,15)(13,20,22,16)(19,24,21,23)(25,28,30,26)(27,33,35,31)(29,36,37,32)(34,39,40,38);(1,4,6,3)(2,8,7,5)(9,13)(10,16)(11,19)(12,20)(14,22)(15,23)(17,21)(18,24);(1,5,6,8)(2,3,7,4)(9,11)(10,15)(12,18)(13,19)(14,17)(16,23)(20,24)(21,22)(25,29,30,37)(26,32,28,36)(27,34,35,40)(31,38,33,39)Polycyclic presentation:
2^9|a^2=gi,b^2=gh,c^2=gh,d^2=hi,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8118872);