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 | |
4270 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,16,20)(11,21,23,17)(12,13,14,18)(15,19,22,24)(26,28)(31,33)(32,36)(38,39);(1,2)(3,7)(4,5)(6,8)(9,11)(10,17)(12,19)(13,15)(14,24)(16,23)(18,22)(20,21)(25,26,30,28)(27,33,35,31)(29,32,37,36)(34,39,40,38);(2,5)(6,7)(9,12,16,14)(10,18,20,13)(11,19,23,24)(15,17,22,21)(25,27,30,35)(26,31,28,33)(29,34,37,40)(32,38,36,39);(9,13)(10,12)(11,22)(14,20)(15,23)(16,18)(17,24)(19,21)(25,28,30,26)(27,33,35,31)(29,36,37,32)(34,39,40,38);(1,3)(2,6)(4,8)(5,7)(9,14,16,12)(10,13,20,18)(11,19,23,24)(15,17,22,21);(1,2)(3,6)(4,5)(7,8)(9,15,16,22)(10,19,20,24)(11,13,23,18)(12,21,14,17)(25,29,30,37)(26,32,28,36)(27,34,35,40)(31,38,33,39)Polycyclic presentation:
2^9|a^2=h,b^2=gh,c^2=g,d^2=gh,e^2=h,f^2=g,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, 8106399);