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 | |
4345 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,4)(6,7)(9,10,14,15)(11,16,12,19)(13,17,21,22)(18,23,20,24)(26,28)(29,37)(31,33)(34,38);(1,2,5,4)(3,7,8,6)(10,15)(16,19)(17,22)(23,24)(26,28)(31,33)(32,36)(39,40);(1,3,5,8)(2,6,4,7)(9,11,14,12)(10,16,15,19)(13,18,21,20)(17,23,22,24)(25,26,30,28)(27,31,35,33)(29,32,37,36)(34,39,38,40);(1,4,5,2)(3,7,8,6)(9,12,14,11)(10,16,15,19)(13,18,21,20)(17,24,22,23)(25,27,30,35)(26,31,28,33)(29,38,37,34)(32,40,36,39);(9,10)(11,16)(12,19)(13,17)(14,15)(18,23)(20,24)(21,22)(25,28,30,26)(27,33,35,31)(29,36,37,32)(34,40,38,39);(9,13)(10,17)(11,18)(12,20)(14,21)(15,22)(16,23)(19,24)(25,29,30,37)(26,32,28,36)(27,34,35,38)(31,39,33,40)Polycyclic presentation:
2^9|a^2=ghi,b^2=hi,c^2=gh,d^2=gh,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=egi,e^d=eghi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10330042);