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 | |
4744 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,5)(6,8)(9,10,14,18)(11,19,21,15)(12,22,20,16)(13,17,23,24)(25,26,31,35)(27,30,34,39)(28,38,37,32)(29,33,36,40);(3,7)(6,8)(9,11,14,21)(10,15,18,19)(12,13,20,23)(16,17,22,24)(25,27,31,34)(26,30,35,39)(28,29,37,36)(32,40,38,33);(1,2)(3,6)(4,5)(7,8)(25,28,31,37)(26,32,35,38)(27,36,34,29)(30,33,39,40);(1,3)(2,6)(4,7)(5,8)(9,12)(10,16)(11,13)(14,20)(15,17)(18,22)(19,24)(21,23)(25,27)(26,30)(28,36)(29,37)(31,34)(32,33)(35,39)(38,40);(1,2)(3,6)(4,5)(7,8)(9,10)(11,15)(12,16)(13,17)(14,18)(19,21)(20,22)(23,24)(25,29,31,36)(26,33,35,40)(27,37,34,28)(30,32,39,38);(9,13)(10,17)(11,20)(12,21)(14,23)(15,22)(16,19)(18,24)(25,30)(26,34)(27,35)(28,33)(29,32)(31,39)(36,38)(37,40)Polycyclic presentation:
2^9|a^2=gi,b^2=gi,c^2=i,e^2=i,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8362952);