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 | #34976 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4438 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,6)(7,8)(9,10)(11,19)(12,14)(13,16)(15,18)(17,23)(20,24)(21,22)(25,26)(27,37)(28,29)(30,34)(31,35)(32,36)(33,40)(38,39);(1,2)(3,7)(4,8)(5,6)(25,27)(26,33)(28,35)(29,39)(30,38)(31,34)(32,40)(36,37);(9,11,15,13)(10,16,18,19)(12,20,23,22)(14,21,17,24)(25,28,32,30)(26,29,36,34)(27,38,40,35)(31,37,39,33);(9,12)(10,17)(11,20)(13,22)(14,18)(15,23)(16,24)(19,21)(25,29,32,34)(26,28,36,30)(27,39,40,31)(33,35,37,38);(1,3,5,4)(2,7,6,8)(9,13,15,11)(10,16,18,19)(12,22,23,20)(14,21,17,24)(25,30,32,28)(26,34,36,29)(27,38,40,35)(31,37,39,33);(1,4,5,3)(2,8,6,7)(9,14,15,17)(10,12,18,23)(11,21,13,24)(16,20,19,22)(25,31,32,39)(26,35,36,38)(27,34,40,29)(28,37,30,33)Polycyclic presentation:
2^9|c^2=hi,d^2=i,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=dh,c^b=ci,f^e=fhi104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7562717);