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 | |
4524 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)(11,13)(12,16)(14,15)(17,18)(19,23)(20,21)(22,24)(25,26,30,32)(27,33,35,28)(29,31,38,37)(34,39,40,36);(1,3,4,7)(2,5,6,8)(9,11,14,18)(10,13,15,17)(12,20,22,19)(16,21,24,23)(25,27,30,35)(26,28,32,33)(29,34,38,40)(31,36,37,39);(1,2)(3,5)(4,6)(7,8)(10,15)(11,18)(16,24)(19,20)(25,28,30,33)(26,27,32,35)(29,36,38,39)(31,34,37,40);(1,3)(2,5)(4,7)(6,8)(9,12)(10,16)(11,19)(13,23)(14,22)(15,24)(17,21)(18,20)(25,26)(27,28)(29,37)(30,32)(31,38)(33,35)(34,39)(36,40);(1,2)(3,5)(4,6)(7,8)(9,11)(10,17)(12,19)(13,15)(14,18)(16,21)(20,22)(23,24)(25,28)(26,27)(29,36)(30,33)(31,34)(32,35)(37,40)(38,39);(9,13,14,17)(10,11,15,18)(12,21,22,23)(16,20,24,19)(25,29,30,38)(26,31,32,37)(27,34,35,40)(28,36,33,39)Polycyclic presentation:
2^9|a^2=ghi,b^2=gh,c^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8541287);