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 | |
4271 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,15,16)(11,19,13,17)(12,21,22,18)(14,20,24,23)(25,26)(27,28)(29,33)(30,31)(32,36)(34,35)(37,38)(39,40);(1,2)(3,6)(4,5)(7,8)(10,16)(12,22)(17,19)(20,23)(26,31)(27,32)(33,38)(34,39);(2,5)(6,8)(9,11,15,13)(10,17,16,19)(12,20,22,23)(14,18,24,21)(25,27,30,32)(26,28,31,36)(29,34,37,39)(33,35,38,40);(3,7)(6,8)(9,12)(10,18)(11,20)(13,23)(14,19)(15,22)(16,21)(17,24)(25,28)(26,32)(27,31)(29,35)(30,36)(33,39)(34,38)(37,40);(1,3)(2,6)(4,7)(5,8)(9,13,15,11)(10,19,16,17)(12,20,22,23)(14,18,24,21);(1,2)(3,6)(4,5)(7,8)(9,14,15,24)(10,20,16,23)(11,21,13,18)(12,19,22,17)(25,29,30,37)(26,33,31,38)(27,34,32,39)(28,35,36,40)Polycyclic presentation:
2^9|a^2=h,c^2=g,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8180621);