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 | |
4643 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)(11,18)(12,20)(13,15)(14,17)(16,21)(19,23)(22,24)(25,26)(27,30)(28,33)(29,32)(31,36)(34,37)(35,39)(38,40);(1,2)(3,6)(4,7)(5,8)(9,11,14,13)(10,15,17,18)(12,19,21,22)(16,23,20,24)(26,27)(31,33)(32,34)(38,39);(1,3)(2,5)(4,8)(6,7)(12,21)(16,20)(19,22)(23,24)(25,26,30,27)(28,31,36,33)(29,32,37,34)(35,38,40,39);(3,8)(5,6)(9,12)(10,16)(11,19)(13,22)(14,21)(15,23)(17,20)(18,24)(25,27,30,26)(28,33,36,31)(29,34,37,32)(35,39,40,38);(1,2)(3,5)(4,7)(6,8)(9,10)(11,15)(12,16)(13,18)(14,17)(19,23)(20,21)(22,24)(25,28)(26,31)(27,33)(29,35)(30,36)(32,38)(34,39)(37,40);(1,3)(2,6)(4,8)(5,7)(9,13,14,11)(10,15,17,18)(12,19,21,22)(16,24,20,23)(25,29,30,37)(26,32,27,34)(28,35,36,40)(31,38,33,39)Polycyclic presentation:
2^9|b^2=gh,c^2=h,d^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,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, 8609832);