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 | |
4514 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,13,16)(11,17,19,14)(12,15,20,22)(18,23,24,21)(25,26)(27,34)(28,36)(29,31)(30,33)(32,37)(35,39)(38,40);(1,2)(3,7)(4,6)(5,8)(9,11,13,19)(10,14,16,17)(12,18,20,24)(15,21,22,23)(25,26)(27,34)(28,32)(29,31)(30,33)(35,40)(36,37)(38,39);(1,3)(2,5)(4,8)(6,7)(9,11)(10,14)(12,18)(13,19)(15,21)(16,17)(20,24)(22,23)(25,27,30,29)(26,31,33,34)(28,35,37,38)(32,39,36,40);(2,6)(5,7)(9,11)(10,14)(12,18)(13,19)(15,21)(16,17)(20,24)(22,23)(25,26)(27,31)(28,32)(29,34)(30,33)(35,39)(36,37)(38,40);(1,3)(2,7)(4,8)(5,6)(9,10)(11,14)(12,15)(13,16)(17,19)(18,21)(20,22)(23,24)(25,28)(26,32)(27,35)(29,38)(30,37)(31,39)(33,36)(34,40);(1,2)(3,5)(4,6)(7,8)(9,12,13,20)(10,15,16,22)(11,18,19,24)(14,21,17,23)(25,29,30,27)(26,31,33,34)(28,35,37,38)(32,40,36,39)Polycyclic presentation:
2^9|a^2=g,b^2=g,c^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,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, 8597204);