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 | C22×D4 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4163 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,6,5)(3,8,4,7)(9,10,14,18)(11,15,16,12)(13,17,22,21)(19,23,24,20)(25,26,30,34)(27,29,33,31)(28,36,39,32)(35,38,37,40);(3,4)(7,8)(9,10)(11,15)(12,16)(13,17)(14,18)(19,23)(20,24)(21,22)(25,27,30,33)(26,31,34,29)(28,37,39,35)(32,40,36,38);(1,3,6,4)(2,7,5,8)(9,11,14,16)(10,15,18,12)(13,19,22,24)(17,23,21,20)(25,26)(27,31)(28,32)(29,33)(30,34)(35,38)(36,39)(37,40);(1,4,6,3)(2,7,5,8)(9,10)(11,15)(12,16)(13,21)(14,18)(17,22)(19,20)(23,24)(25,28,30,39)(26,32,34,36)(27,35,33,37)(29,40,31,38);(9,12,14,15)(10,16,18,11)(13,20,22,23)(17,24,21,19);(1,5,6,2)(3,8,4,7)(9,13,14,22)(10,17,18,21)(11,19,16,24)(12,20,15,23)(25,29,30,31)(26,33,34,27)(28,38,39,40)(32,35,36,37)Polycyclic presentation:
2^9|a^2=gi,b^2=hi,c^2=gh,d^2=h,e^2=ghi,f^2=gi,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, 8370442);