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 | |
4464 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)(11,16)(12,14)(13,17)(15,19)(18,23)(20,24)(21,22)(25,26,30,33)(27,34,28,31)(29,37,38,32)(35,39,36,40);(1,2)(3,7)(4,5)(6,8)(9,11,15,23)(10,16,19,18)(12,21,13,20)(14,22,17,24)(25,26)(27,34)(28,31)(29,32)(30,33)(35,40)(36,39)(37,38);(2,5)(6,7)(9,12,15,13)(10,17,19,14)(11,20,23,21)(16,22,18,24)(25,27,30,28)(26,31,33,34)(29,35,38,36)(32,39,37,40);(2,5)(6,7)(9,13,15,12)(10,17,19,14)(11,21,23,20)(16,22,18,24)(25,26)(27,31)(28,34)(29,32)(30,33)(35,39)(36,40)(37,38);(1,2)(3,6)(4,5)(7,8)(9,14,15,17)(10,12,19,13)(11,22,23,24)(16,21,18,20)(25,28,30,27)(26,31,33,34)(29,36,38,35)(32,39,37,40);(1,3)(2,6)(4,8)(5,7)(9,11)(10,18)(12,20)(13,21)(14,22)(15,23)(16,19)(17,24)(25,29,30,38)(26,32,33,37)(27,35,28,36)(31,39,34,40)Polycyclic presentation:
2^9|a^2=g,b^2=gh,c^2=h,d^2=gh,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,f^b=fgi,e^c=egi,e^d=eghi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10229570);