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 | |
4544 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)(3,7)(4,5)(6,8)(9,10)(11,15)(12,16)(13,14)(17,18)(19,22)(20,21)(23,24)(25,26)(27,31)(28,29)(30,34)(32,37)(33,40)(35,36)(38,39);(2,5)(6,7)(10,14)(15,18)(16,21)(22,24)(25,27)(26,31)(28,35)(29,36)(30,37)(32,34)(33,38)(39,40);(1,2)(3,6)(4,5)(7,8)(11,17)(12,20)(15,18)(16,21)(25,27,30,37)(26,32,34,31)(28,35,33,38)(29,39,40,36);(1,3,4,8)(2,6,5,7)(9,11,13,17)(10,15,14,18)(12,19,20,23)(16,22,21,24)(25,27)(26,32)(28,38)(29,36)(30,37)(31,34)(33,35)(39,40);(9,10)(11,18)(12,16)(13,14)(15,17)(19,24)(20,21)(22,23)(25,28,30,33)(26,29,34,40)(27,35,37,38)(31,36,32,39);(9,12)(10,16)(11,19)(13,20)(14,21)(15,22)(17,23)(18,24)(25,29)(26,33)(27,36)(28,34)(30,40)(31,38)(32,35)(37,39)Polycyclic presentation:
2^9|c^2=h,d^2=g,e^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,f^c=fgi,e^d=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8834899);