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 | |
4513 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10)(11,18)(12,13)(14,15)(16,19)(17,22)(20,24)(21,23)(26,30)(31,34)(32,36)(38,40);(1,2)(3,7)(4,5)(6,8)(10,15)(11,19)(13,22)(20,23)(25,26,29,30)(27,31,33,34)(28,36,37,32)(35,40,39,38);(2,5)(6,7)(10,15)(12,17)(16,18)(20,23)(27,33)(31,34)(35,39)(38,40);(2,5)(6,7)(9,11,14,19)(10,16,15,18)(12,20,17,23)(13,21,22,24)(25,26)(27,34)(28,32)(29,30)(31,33)(35,40)(36,37)(38,39);(1,2)(3,6)(4,5)(7,8)(9,12,14,17)(10,13,15,22)(11,20,19,23)(16,21,18,24)(25,27)(26,31)(28,35)(29,33)(30,34)(32,38)(36,40)(37,39);(1,3)(2,6)(4,8)(5,7)(9,13)(10,17)(11,21)(12,15)(14,22)(16,23)(18,20)(19,24)(25,28,29,37)(26,32,30,36)(27,35,33,39)(31,38,34,40)Polycyclic presentation:
2^9|b^2=gh,d^2=h,e^2=h,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, 10231326);