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 | |
4380 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2)(3,7)(4,5)(6,8)(9,10)(11,14)(12,17)(13,18)(15,19)(16,21)(20,22)(23,24)(25,26)(27,34)(28,29)(30,31)(32,35)(33,38)(36,40)(37,39);(2,5)(6,7)(10,11)(12,19)(16,18)(20,24)(26,31)(28,33)(32,34)(36,39);(1,2)(3,6)(4,5)(7,8)(10,11)(15,17)(16,18)(22,23)(27,35)(32,34)(36,39)(37,40);(1,3)(2,6)(4,8)(5,7)(9,11,14,10)(12,17,19,15)(13,18,21,16)(20,23,24,22)(25,27,30,35)(26,32,31,34)(28,36,33,39)(29,37,38,40);(9,12,14,19)(10,15,11,17)(13,20,21,24)(16,22,18,23)(25,28,30,33)(26,29,31,38)(27,36,35,39)(32,37,34,40);(9,13,14,21)(10,16,11,18)(12,20,19,24)(15,22,17,23)(25,29)(26,33)(27,37)(28,31)(30,38)(32,39)(34,36)(35,40)Polycyclic presentation:
2^9|d^2=gi,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^b=egi,d^c=dgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8021596);