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 | |
4545 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)(12,20)(16,21)(18,24)(22,23)(25,26,31,32)(27,33,28,29)(30,38,39,34)(35,37,36,40);(2,5)(3,7)(9,10)(11,17)(12,16)(13,14)(15,19)(18,23)(20,21)(22,24)(25,27)(26,29)(28,31)(30,35)(32,33)(34,37)(36,39)(38,40);(1,2,4,5)(3,6,7,8)(10,14)(15,17)(16,21)(22,23)(26,32)(27,28)(34,38)(35,36);(1,3)(2,6)(4,7)(5,8)(9,10)(11,15)(12,16)(13,14)(17,19)(18,22)(20,21)(23,24)(25,28,31,27)(26,33,32,29)(30,36,39,35)(34,40,38,37);(9,11,13,19)(10,15,14,17)(12,18,20,24)(16,22,21,23)(25,29,31,33)(26,27,32,28)(30,37,39,40)(34,35,38,36);(9,12)(10,16)(11,18)(13,20)(14,21)(15,22)(17,23)(19,24)(25,30,31,39)(26,34,32,38)(27,35,28,36)(29,37,33,40)Polycyclic presentation:
2^9|a^2=g,c^2=gh,d^2=g,e^2=ghi,f^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, 8000756);