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 | #34636 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4241 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10)(11,15)(12,19)(13,16)(14,17)(18,23)(20,21)(22,24)(25,26)(27,28)(29,32)(30,37)(31,33)(34,39)(35,40)(36,38);(1,2)(3,7)(4,5)(6,8)(9,10,14,17)(11,15,19,12)(13,16,22,24)(18,23,21,20)(25,27,32,31)(26,33,29,28)(30,35,39,38)(34,40,37,36);(2,5)(6,7)(9,11,14,19)(10,15,17,12)(13,21,22,18)(16,20,24,23)(25,28)(26,31)(27,29)(30,36)(32,33)(34,38)(35,37)(39,40);(3,8)(6,7)(9,12)(10,11)(13,20)(14,15)(16,18)(17,19)(21,24)(22,23)(25,29,32,26)(27,33,31,28)(30,37,39,34)(35,40,38,36);(1,3)(2,6)(4,8)(5,7)(9,10)(11,15)(12,19)(13,16)(14,17)(18,23)(20,21)(22,24)(25,30)(26,34)(27,35)(28,36)(29,37)(31,38)(32,39)(33,40);(1,2)(3,6)(4,5)(7,8)(9,13)(10,16)(11,18)(12,20)(14,22)(15,23)(17,24)(19,21)(25,31,32,27)(26,33,29,28)(30,38,39,35)(34,40,37,36)Polycyclic presentation:
2^9|b^2=g,c^2=gh,d^2=h,f^2=h,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, 8255474);