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 | #53057 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4301 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,14,18)(11,13,17,15)(12,20,22,16)(19,23,24,21)(25,26,31,32)(27,33,28,29)(30,34,39,40)(35,38,36,37);(1,2)(3,6)(4,7)(5,8)(9,11)(10,15)(12,19)(13,18)(14,17)(16,23)(20,21)(22,24)(26,32)(27,28)(34,40)(35,36);(1,3)(2,5)(4,8)(6,7)(25,27,31,28)(26,29,32,33)(30,35,39,36)(34,37,40,38);(9,12,14,22)(10,16,18,20)(11,19,17,24)(13,21,15,23)(25,28,31,27)(26,33,32,29)(30,36,39,35)(34,38,40,37);(1,3)(2,6)(4,8)(5,7)(9,11)(10,13)(12,19)(14,17)(15,18)(16,21)(20,23)(22,24)(25,29,31,33)(26,27,32,28)(30,38,39,37)(34,36,40,35);(1,2)(3,5)(4,7)(6,8)(9,13)(10,17)(11,18)(12,21)(14,15)(16,24)(19,20)(22,23)(25,30,31,39)(26,34,32,40)(27,35,28,36)(29,37,33,38)Polycyclic presentation:
2^9|a^2=g,c^2=h,d^2=g,e^2=h,f^2=h,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, 8003462);