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 | |
4619 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,14,13)(11,17,19,15)(12,20,22,16)(18,23,21,24)(25,26)(27,34)(28,31)(29,32)(30,33)(35,37)(36,39)(38,40);(1,2)(3,7)(4,5)(6,8)(9,11,14,19)(10,15,13,17)(12,21,22,18)(16,24,20,23)(25,27)(26,28)(29,35)(30,31)(32,36)(33,34)(37,40)(38,39);(2,5)(6,7)(10,13)(15,17)(16,20)(23,24)(25,28,30,34)(26,31,33,27)(29,37,38,36)(32,35,40,39);(3,8)(6,7)(9,11)(10,15)(12,21)(13,17)(14,19)(16,24)(18,22)(20,23)(25,27)(26,28)(29,35)(30,31)(32,36)(33,34)(37,40)(38,39);(1,3)(2,6)(4,8)(5,7)(9,12)(10,16)(11,18)(13,20)(14,22)(15,23)(17,24)(19,21)(25,26)(27,34)(28,31)(29,32)(30,33)(35,37)(36,39)(38,40);(1,2)(3,6)(4,5)(7,8)(9,13,14,10)(11,17,19,15)(12,20,22,16)(18,24,21,23)(25,29,30,38)(26,32,33,40)(27,35,31,39)(28,36,34,37)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=gh,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, 8254069);