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 | #34990 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4591 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)(3,6,8,4)(9,10)(11,17)(12,19)(13,16)(14,15)(18,24)(20,21)(22,23)(25,26)(27,32)(28,34)(29,33)(30,31)(35,39)(36,40)(37,38);(1,3,5,8)(2,6,7,4)(9,11,14,19)(10,12,15,17)(13,18,22,21)(16,20,23,24)(26,31)(32,34)(33,38)(39,40);(2,7)(3,8)(10,15)(12,17)(16,23)(20,24)(25,27,30,28)(26,32,31,34)(29,35,37,36)(33,39,38,40);(1,4,5,6)(2,3,7,8)(9,11)(10,12)(13,21)(14,19)(15,17)(16,24)(18,22)(20,23)(25,28,30,27)(26,32,31,34)(29,35,37,36)(33,40,38,39);(1,2)(3,4)(5,7)(6,8)(9,12)(10,11)(13,20)(14,17)(15,19)(16,18)(21,23)(22,24)(25,26)(27,34)(28,32)(29,33)(30,31)(35,40)(36,39)(37,38);(1,4)(2,3)(5,6)(7,8)(9,13)(10,16)(11,18)(12,20)(14,22)(15,23)(17,24)(19,21)(25,29)(26,33)(27,35)(28,36)(30,37)(31,38)(32,39)(34,40)Polycyclic presentation:
2^9|a^2=i,b^2=h,c^2=ghi,d^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8510025);