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 | #2669 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4449 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)(3,5)(4,7)(6,8)(9,10)(11,18)(12,20)(13,16)(14,17)(15,22)(19,21)(23,24);(1,3,4,8)(2,5,7,6)(9,11,14,16)(10,13,17,18)(12,19,22,23)(15,21,20,24)(26,30)(31,37)(32,38)(39,40);(1,3)(2,6)(4,8)(5,7)(25,26)(27,31)(28,32)(29,30)(33,39)(34,37)(35,40)(36,38);(1,2)(3,6)(4,7)(5,8)(9,12)(10,15)(11,19)(13,21)(14,22)(16,23)(17,20)(18,24);(9,10)(11,13)(12,15)(14,17)(16,18)(19,21)(20,22)(23,24)(25,27,29,34)(26,31,30,37)(28,35,36,33)(32,40,38,39);(9,13)(10,16)(11,17)(12,21)(14,18)(15,23)(19,20)(22,24)(25,28,29,36)(26,32,30,38)(27,33,34,35)(31,39,37,40)Polycyclic presentation:
2^9|b^2=gh,e^2=hi,f^2=hi,b^a=bg,c^a=ch,d^a=dg,c^b=ci,d^b=dh,f^e=fghi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8003362);