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 | |
4413 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)(9,10,16,19)(11,20,22,17)(12,15,13,23)(14,24,21,18)(25,26)(27,33)(28,30)(29,32)(31,35)(34,39)(36,37)(38,40);(1,2,5,4)(3,7,8,6)(9,11)(10,17)(12,14)(13,21)(15,18)(16,22)(19,20)(23,24)(26,28)(31,33)(32,36)(38,39);(1,3,5,8)(2,6,4,7)(9,12,16,13)(10,15,19,23)(11,14,22,21)(17,18,20,24)(25,27,30,35)(26,31,28,33)(29,34,37,40)(32,38,36,39);(1,4,5,2)(3,7,8,6)(9,13,16,12)(10,15,19,23)(11,21,22,14)(17,18,20,24)(25,28,30,26)(27,33,35,31)(29,32,37,36)(34,38,40,39);(9,14,16,21)(10,18,19,24)(11,12,22,13)(15,20,23,17);(9,15)(10,13)(11,18)(12,19)(14,20)(16,23)(17,21)(22,24)(25,29)(26,32)(27,34)(28,36)(30,37)(31,38)(33,39)(35,40)Polycyclic presentation:
2^9|a^2=ghi,b^2=i,c^2=g,d^2=g,e^2=ghi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8353290);