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 | #59163 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4167 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)(11,19)(12,21)(13,18)(14,16)(15,22)(17,23)(20,24)(25,26,31,35)(27,34,30,32)(28,29,33,40)(36,39,38,37);(1,2)(3,7)(4,5)(6,8)(9,11,14,22)(10,15,16,19)(12,18,23,20)(13,21,24,17)(25,26,31,35)(27,32,30,34)(28,29,33,40)(36,37,38,39);(2,5)(6,7)(10,16)(13,24)(15,19)(17,21)(25,27,31,30)(26,32,35,34)(28,36,33,38)(29,37,40,39);(2,5)(6,7)(9,12,14,23)(10,17,16,21)(11,20,22,18)(13,15,24,19)(25,28)(26,29)(27,36)(30,38)(31,33)(32,37)(34,39)(35,40);(1,2)(3,6)(4,5)(7,8)(9,13,14,24)(10,18,16,20)(11,21,22,17)(12,15,23,19)(25,29)(26,33)(27,37)(28,35)(30,39)(31,40)(32,38)(34,36);(1,3)(2,6)(4,8)(5,7)(9,11)(10,15)(12,20)(13,21)(14,22)(16,19)(17,24)(18,23)(25,30,31,27)(26,34,35,32)(28,38,33,36)(29,39,40,37)Polycyclic presentation:
2^9|a^2=gh,b^2=h,c^2=gh,d^2=g,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,f^b=fgi,e^c=egi,e^d=eghi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10230155);