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 | |
4426 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2)(3,5)(4,8)(6,7)(9,10)(11,14)(12,21)(13,17)(15,18)(16,20)(19,24)(22,23)(25,26,30,32)(27,31,35,40)(28,33,36,37)(29,38,39,34);(2,7)(3,8)(9,11)(10,14)(12,13)(15,19)(16,22)(17,21)(18,24)(20,23)(25,27,30,35)(26,31,32,40)(28,29,36,39)(33,38,37,34);(1,3,6,8)(2,4,7,5)(11,19)(12,20)(14,24)(16,21)(26,32)(27,35)(29,39)(33,37);(1,4,6,5)(2,3,7,8)(9,12)(10,16)(11,13)(14,22)(15,20)(17,24)(18,21)(19,23)(25,26)(27,31)(28,37)(29,34)(30,32)(33,36)(35,40)(38,39);(1,2)(3,4)(5,8)(6,7)(9,13)(10,17)(11,20)(12,19)(14,16)(15,23)(18,22)(21,24)(25,28,30,36)(26,33,32,37)(27,29,35,39)(31,38,40,34);(1,5,6,4)(2,8,7,3)(9,14)(10,11)(12,22)(13,16)(15,24)(17,20)(18,19)(21,23)(25,29)(26,34)(27,36)(28,35)(30,39)(31,33)(32,38)(37,40)Polycyclic presentation:
2^9|a^2=gh,b^2=gh,c^2=g,d^2=g,e^2=gh,f^2=g,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, 8834894);