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 | |
4519 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10)(11,19)(12,17)(13,21)(14,15)(16,20)(18,23)(22,24)(25,26,30,34)(27,31,37,32)(28,33,39,29)(35,40,38,36);(1,2)(3,7)(4,5)(6,8)(10,15)(11,20)(17,24)(18,21)(25,26)(27,31)(28,29)(30,34)(32,37)(33,39)(35,36)(38,40);(2,5)(3,8)(10,15)(11,20)(12,22)(13,23)(25,27,30,37)(26,31,34,32)(28,38,39,35)(29,40,33,36);(1,3,4,8)(2,6,5,7)(9,11)(10,16)(12,18)(13,17)(14,20)(15,19)(21,22)(23,24)(25,27)(26,32)(28,38)(29,36)(30,37)(31,34)(33,40)(35,39);(1,3)(2,6)(4,8)(5,7)(9,12)(10,17)(11,21)(13,19)(14,22)(15,24)(16,23)(18,20)(25,28)(26,33)(27,35)(29,34)(30,39)(31,40)(32,36)(37,38);(1,2)(3,6)(4,5)(7,8)(9,13,14,23)(10,18,15,21)(11,17,20,24)(12,19,22,16)(25,29,30,33)(26,28,34,39)(27,36,37,40)(31,35,32,38)Polycyclic presentation:
2^9|a^2=gh,c^2=gh,d^2=i,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8969419);