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 | |
4334 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,15,12)(11,16,13,17)(14,18,22,20)(19,23,21,24)(25,26)(27,35)(28,38)(29,40)(30,34)(31,37)(32,39)(33,36);(1,2)(3,6)(4,5)(7,8)(9,11,15,13)(10,16,12,17)(14,19,22,21)(18,23,20,24)(25,27)(26,31)(28,36)(29,39)(30,37)(32,33)(34,35)(38,40);(2,5)(6,8)(9,11,15,13)(10,17,12,16)(14,19,22,21)(18,24,20,23)(25,28,30,39)(26,32,34,38)(27,36,37,29)(31,33,35,40);(3,7)(6,8)(9,12,15,10)(11,16,13,17)(14,20,22,18)(19,23,21,24)(25,29)(26,33)(27,28)(30,36)(31,38)(32,35)(34,40)(37,39);(1,3)(2,6)(4,7)(5,8)(9,13,15,11)(10,16,12,17)(14,21,22,19)(18,23,20,24)(25,27)(26,31)(28,36)(29,39)(30,37)(32,33)(34,35)(38,40);(1,2)(3,6)(4,5)(7,8)(9,14,15,22)(10,18,12,20)(11,19,13,21)(16,23,17,24)(25,26)(27,31)(28,38)(29,33)(30,34)(32,39)(35,37)(36,40)Polycyclic presentation:
2^9|a^2=gh,b^2=gh,c^2=h,d^2=gh,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,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, 8183544);