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 | |
4384 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,6)(3,7,8,4)(9,10,13,16)(11,17,19,14)(12,15,21,20)(18,23,24,22)(25,26)(27,35)(28,32)(29,33)(30,34)(31,36)(37,40)(38,39);(1,3,5,8)(2,4,6,7)(9,10)(11,14)(12,15)(13,16)(17,19)(18,22)(20,21)(23,24)(27,36)(28,38)(31,35)(32,39);(1,2)(3,4)(5,6)(7,8)(25,27,30,36)(26,31,34,35)(28,37,38,29)(32,33,39,40);(1,2)(3,7)(4,8)(5,6)(9,10)(11,14)(12,20)(13,16)(15,21)(17,19)(18,23)(22,24)(25,28)(26,32)(27,37)(29,36)(30,38)(31,33)(34,39)(35,40);(9,11,13,19)(10,14,16,17)(12,18,21,24)(15,22,20,23)(25,29,30,37)(26,33,34,40)(27,28,36,38)(31,39,35,32);(1,4,5,7)(2,3,6,8)(9,12,13,21)(10,15,16,20)(11,18,19,24)(14,22,17,23)(25,26)(27,31)(28,39)(29,33)(30,34)(32,38)(35,36)(37,40)Polycyclic presentation:
2^9|a^2=g,b^2=hi,c^2=i,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8343981);