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 | |
4605 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,15,13)(11,17,12,16)(14,18,22,21)(19,24,20,23)(25,26)(27,30)(28,33)(29,32)(31,36)(34,37)(35,39)(38,40);(1,2)(3,7)(4,5)(6,8)(9,11,15,12)(10,16,13,17)(14,19,22,20)(18,23,21,24)(26,27)(31,33)(32,34)(38,39);(2,5)(6,7)(14,22)(18,21)(19,20)(23,24)(25,26,30,27)(28,31,36,33)(29,32,37,34)(35,38,40,39);(3,8)(6,7)(9,12,15,11)(10,17,13,16)(14,20,22,19)(18,24,21,23)(25,27,30,26)(28,33,36,31)(29,34,37,32)(35,39,40,38);(1,3)(2,6)(4,8)(5,7)(9,13,15,10)(11,16,12,17)(14,21,22,18)(19,23,20,24)(25,28,30,36)(26,31,27,33)(29,35,37,40)(32,38,34,39);(1,2)(3,6)(4,5)(7,8)(9,14,15,22)(10,18,13,21)(11,19,12,20)(16,23,17,24)(25,29,30,37)(26,32,27,34)(28,35,36,40)(31,38,33,39)Polycyclic presentation:
2^9|a^2=gh,b^2=gh,c^2=h,d^2=g,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8254092);