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 | #52586 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4294 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)(11,12)(13,21)(14,16)(15,18)(17,22)(19,23)(20,24)(26,27)(28,35)(32,34)(36,40);(2,5)(3,7)(9,11,14,18)(10,15,16,12)(13,19,22,24)(17,23,21,20)(25,26,30,27)(28,33,35,31)(29,32,37,34)(36,39,40,38);(1,2)(3,6)(4,5)(7,8)(10,16)(11,18)(17,21)(19,24)(25,27,30,26)(28,33,35,31)(29,34,37,32)(36,39,40,38);(1,3)(2,6)(4,7)(5,8)(9,11)(10,15)(12,16)(13,19)(14,18)(17,23)(20,21)(22,24)(25,28,30,35)(26,31,27,33)(29,36,37,40)(32,38,34,39);(9,12,14,15)(10,11,16,18)(13,20,22,23)(17,19,21,24)(25,29,30,37)(26,32,27,34)(28,36,35,40)(31,38,33,39);(9,13)(10,17)(11,19)(12,20)(14,22)(15,23)(16,21)(18,24)Polycyclic presentation:
2^9|b^2=g,c^2=hi,d^2=hi,e^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fghi,c^b=ci,d^b=dh,e^b=eghi,d^c=dghi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8000677);