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 | #44065 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4298 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)(3,5)(4,7)(6,8)(9,10,15,13)(11,19,21,16)(12,20,22,17)(14,18,24,23);(1,2,6,8)(3,5,7,4)(9,11)(10,16)(12,18)(13,19)(14,20)(15,21)(17,24)(22,23)(25,26)(27,28)(29,32)(30,33)(31,35)(34,36)(37,38)(39,40);(1,3,6,7)(2,4,8,5)(25,27)(26,31)(28,33)(29,34)(30,35)(32,37)(36,40)(38,39);(1,4,6,5)(2,3,8,7)(9,12,15,22)(10,17,13,20)(11,18,21,23)(14,16,24,19)(25,28,30,31)(26,27,33,35)(29,37,39,36)(32,38,40,34);(1,2)(3,4)(5,7)(6,8)(9,13,15,10)(11,19,21,16)(12,17,22,20)(14,18,24,23)(25,27)(26,28)(29,38)(30,35)(31,33)(32,37)(34,39)(36,40);(1,5,6,4)(2,7,8,3)(9,14,15,24)(10,18,13,23)(11,20,21,17)(12,19,22,16)(25,29,30,39)(26,32,33,40)(27,34,35,38)(28,36,31,37)Polycyclic presentation:
2^9|a^2=gh,b^2=h,c^2=h,d^2=ghi,e^2=gh,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^d=egi,f^d=fgi,f^e=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9139998);