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 | #51940 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4621 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,7)(3,4,8,6)(9,10,14,15)(11,19,20,16)(12,17,22,21)(13,18,23,24)(25,26,30,31)(27,33,35,28)(29,37,38,32)(34,36,40,39);(1,3,5,8)(2,6,7,4)(10,15)(13,23)(16,19)(17,21)(26,31)(28,33)(29,38)(34,40);(1,2)(3,4)(5,7)(6,8)(11,20)(12,22)(16,19)(17,21)(25,27,30,35)(26,28,31,33)(29,34,38,40)(32,36,37,39);(1,4)(2,3)(5,6)(7,8)(9,11,14,20)(10,16,15,19)(12,18,22,24)(13,21,23,17)(25,27)(26,28)(29,34)(30,35)(31,33)(32,36)(37,39)(38,40);(1,3)(2,4)(5,8)(6,7)(9,12,14,22)(10,17,15,21)(11,18,20,24)(13,19,23,16)(25,28)(26,27)(29,36)(30,33)(31,35)(32,34)(37,40)(38,39);(1,4)(2,3)(5,6)(7,8)(9,13)(10,18)(11,21)(12,19)(14,23)(15,24)(16,22)(17,20)(25,29,30,38)(26,32,31,37)(27,34,35,40)(28,36,33,39)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=hi,d^2=gh,e^2=gh,f^2=hi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8721205);