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 | |
4297 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,15,17)(11,16,19,14)(12,13,18,24)(20,21,23,22)(26,31)(27,34)(29,37)(36,39);(2,3)(6,7)(9,11)(10,16)(12,20)(13,21)(14,17)(15,19)(18,23)(22,24)(25,26,30,31)(27,28,34,32)(29,33,37,38)(35,36,40,39);(1,2,5,3)(4,6,8,7)(10,17)(11,19)(12,18)(21,22)(25,27,30,34)(26,32,31,28)(29,35,37,40)(33,39,38,36);(9,12,15,18)(10,13,17,24)(11,20,19,23)(14,22,16,21)(25,27)(26,28)(29,35)(30,34)(31,32)(33,36)(37,40)(38,39);(1,3,5,2)(4,6,8,7)(9,13)(10,18)(11,21)(12,17)(14,20)(15,24)(16,23)(19,22)(25,28)(26,27)(29,36)(30,32)(31,34)(33,35)(37,39)(38,40);(1,4)(2,6)(3,7)(5,8)(9,14)(10,19)(11,17)(12,22)(13,23)(15,16)(18,21)(20,24)(25,29,30,37)(26,33,31,38)(27,35,34,40)(28,36,32,39)Polycyclic presentation:
2^9|a^2=gh,b^2=g,c^2=ghi,d^2=gh,e^2=hi,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9756178);