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 | |
4448 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,4,6)(3,5,7,8)(9,10)(11,15)(12,20)(13,14)(16,22)(17,19)(18,24)(21,23)(25,26,30,33)(27,31,35,28)(29,32,38,40)(34,37,39,36);(3,7)(5,8)(9,10)(11,17)(12,20)(13,14)(15,19)(16,22)(18,23)(21,24)(25,27,30,35)(26,28,33,31)(29,34,38,39)(32,36,40,37);(1,2)(3,5)(4,6)(7,8)(10,14)(15,17)(16,20)(23,24)(25,28,30,31)(26,27,33,35)(29,37,38,36)(32,39,40,34);(1,3)(2,5)(4,7)(6,8)(9,11)(10,15)(12,21)(13,19)(14,17)(16,24)(18,22)(20,23)(25,26)(27,28)(29,32)(30,33)(31,35)(34,36)(37,39)(38,40);(1,2)(3,5)(4,6)(7,8)(9,12)(10,16)(11,18)(13,22)(14,20)(15,23)(17,24)(19,21)(25,27)(26,31)(28,33)(29,34)(30,35)(32,37)(36,40)(38,39);(9,10)(11,15)(12,16)(13,14)(17,19)(18,23)(20,22)(21,24)(25,29,30,38)(26,32,33,40)(27,34,35,39)(28,36,31,37)Polycyclic presentation:
2^9|a^2=ghi,b^2=g,c^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, 8255770);