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 | |
4670 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,4)(6,7)(9,10)(11,18)(12,20)(13,16)(14,17)(15,21)(19,23)(22,24)(25,26)(27,32)(28,29)(30,35)(31,33)(34,40)(36,39)(37,38);(1,2,5,4)(3,7,8,6)(9,11,14,16)(10,13,17,18)(12,19,21,24)(15,23,20,22)(25,27,31,35)(26,32,33,30)(28,38,34,36)(29,37,40,39);(1,3,5,8)(2,6,4,7)(12,21)(15,20)(19,24)(22,23)(26,33)(27,35)(28,34)(37,39);(1,4,5,2)(3,7,8,6)(9,12)(10,15)(11,19)(13,23)(14,21)(16,24)(17,20)(18,22)(25,28,31,34)(26,29,33,40)(27,36,35,38)(30,37,32,39);(9,10)(11,13)(12,15)(14,17)(16,18)(19,23)(20,21)(22,24)(25,29)(26,34)(27,37)(28,33)(30,38)(31,40)(32,36)(35,39);(9,13)(10,16)(11,17)(12,22)(14,18)(15,19)(20,24)(21,23)(25,30,31,32)(26,35,33,27)(28,39,34,37)(29,36,40,38)Polycyclic presentation:
2^9|b^2=gh,c^2=hi,d^2=i,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8626150);