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 |
| prime radical | C2×#120 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4309 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,7)(3,8,6,4)(9,10,15,19)(11,20,21,16)(12,17,22,24)(13,14,18,23)(25,26,29,32)(27,33,35,30)(28,31,37,40)(34,39,36,38);(1,3)(2,4)(5,6)(7,8)(11,21)(12,22)(16,20)(17,24)(25,27,29,35)(26,30,32,33)(28,34,37,36)(31,38,40,39);(9,11)(10,16)(12,18)(13,22)(14,17)(15,21)(19,20)(23,24)(25,26)(27,30)(28,31)(29,32)(33,35)(34,38)(36,39)(37,40);(1,4,5,8)(2,6,7,3)(9,12,15,22)(10,17,19,24)(11,18,21,13)(14,20,23,16)(25,27)(26,30)(28,36)(29,35)(31,39)(32,33)(34,37)(38,40);(1,3)(2,4)(5,6)(7,8)(9,13,15,18)(10,14,19,23)(11,22,21,12)(16,17,20,24);(1,4)(2,3)(5,8)(6,7)(9,14)(10,18)(11,17)(12,16)(13,19)(15,23)(20,22)(21,24)(25,28)(26,31)(27,34)(29,37)(30,38)(32,40)(33,39)(35,36)Polycyclic presentation:
2^9|a^2=gi,b^2=hi,d^2=gh,e^2=i,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, 8353038);