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 |
| prime radical | #53059 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4385 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,6)(3,4,7,8)(9,10,16,22)(11,23,24,17)(12,19,13,18)(14,15,21,20)(25,26,30,29)(27,31,33,35)(28,32,36,37)(34,38,39,40);(1,3)(2,4)(5,7)(6,8)(9,11)(10,17)(12,20)(13,15)(14,18)(16,24)(19,21)(22,23)(27,33)(31,35)(34,39)(38,40);(2,6)(3,7)(9,12,16,13)(10,18,22,19)(11,20,24,15)(14,23,21,17)(25,27,30,33)(26,31,29,35)(28,34,36,39)(32,38,37,40);(3,7)(4,8)(9,13,16,12)(10,19,22,18)(11,20,24,15)(14,23,21,17)(25,28)(26,32)(27,34)(29,37)(30,36)(31,38)(33,39)(35,40);(1,2)(3,8)(4,7)(5,6)(9,14,16,21)(10,20,22,15)(11,19,24,18)(12,17,13,23)(25,27)(26,31)(28,34)(29,35)(30,33)(32,38)(36,39)(37,40);(1,4,5,8)(2,7,6,3)(9,15,16,20)(10,21,22,14)(11,13,24,12)(17,19,23,18)(25,29,30,26)(27,35,33,31)(28,32,36,37)(34,38,39,40)Polycyclic presentation:
2^9|a^2=gi,c^2=hi,d^2=g,e^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, 8546906);