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 | #49822 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4236 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,6)(3,4,7,8)(26,29)(27,28)(34,37)(35,36);(2,6)(3,7)(9,10,14,15)(11,13,17,22)(12,19,21,16)(18,24,23,20)(25,26,31,29)(27,33,28,32)(30,34,38,37)(35,40,36,39);(1,3)(2,4)(5,7)(6,8)(10,15)(13,22)(16,19)(20,24)(25,27,31,28)(26,32,29,33)(30,35,38,36)(34,39,37,40);(1,4)(2,3)(5,8)(6,7)(9,11,14,17)(10,13,15,22)(12,18,21,23)(16,20,19,24)(25,28,31,27)(26,33,29,32)(30,36,38,35)(34,40,37,39);(9,12,14,21)(10,16,15,19)(11,18,17,23)(13,20,22,24)(25,29,31,26)(27,33,28,32)(30,37,38,34)(35,40,36,39);(9,13)(10,17)(11,15)(12,20)(14,22)(16,23)(18,19)(21,24)(25,30,31,38)(26,34,29,37)(27,35,28,36)(32,39,33,40)Polycyclic presentation:
2^9|a^2=gh,b^2=ghi,c^2=h,d^2=ghi,e^2=ghi,f^2=h,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, 8137868);