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 | #59092 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4284 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(3,7)(5,8)(12,19)(16,21)(18,23)(22,24)(26,31)(28,34)(32,40)(35,38);(1,2,4,6)(3,5,7,8)(9,10)(11,15)(12,16)(13,14)(17,20)(18,22)(19,21)(23,24)(25,26)(27,32)(28,33)(29,34)(30,31)(35,39)(36,38)(37,40);(1,3)(2,5)(4,7)(6,8)(10,14)(11,17)(16,21)(18,23)(25,27,30,37)(26,32,31,40)(28,38,34,35)(29,39,33,36);(1,2)(3,5)(4,6)(7,8)(9,11,13,17)(10,15,14,20)(12,18,19,23)(16,22,21,24)(25,28,30,34)(26,33,31,29)(27,35,37,38)(32,39,40,36);(1,3)(2,5)(4,7)(6,8)(9,10)(11,15)(12,16)(13,14)(17,20)(18,22)(19,21)(23,24)(25,29,30,33)(26,34,31,28)(27,36,37,39)(32,38,40,35);(9,12)(10,16)(11,18)(13,19)(14,21)(15,22)(17,23)(20,24)(25,26)(27,32)(28,33)(29,34)(30,31)(35,39)(36,38)(37,40)Polycyclic presentation:
2^9|b^2=h,c^2=g,d^2=gi,e^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,e^c=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8034917);