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 | |
4518 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)(3,7)(4,6)(5,8)(9,10)(11,21)(12,17)(13,18)(14,19)(15,20)(16,22)(23,24)(26,29)(28,32)(33,37)(36,38);(1,2,4,6)(3,5,8,7)(11,22)(12,23)(16,21)(17,24)(25,26,31,29)(27,32,35,28)(30,33,39,37)(34,38,40,36);(1,2)(3,5)(4,6)(7,8)(9,11)(10,16)(12,19)(13,23)(14,24)(15,22)(17,18)(20,21)(25,27,31,35)(26,32,29,28)(30,34,39,40)(33,38,37,36);(1,3)(2,5)(4,8)(6,7)(9,12,15,23)(10,17,20,24)(11,19,22,13)(14,16,18,21)(25,28)(26,27)(29,35)(30,38)(31,32)(33,40)(34,37)(36,39);(9,13,15,19)(10,18,20,14)(11,23,22,12)(16,17,21,24)(25,29,31,26)(27,28,35,32)(30,37,39,33)(34,36,40,38);(9,14,15,18)(10,19,20,13)(11,24,22,17)(12,21,23,16)(25,30,31,39)(26,33,29,37)(27,34,35,40)(28,36,32,38)Polycyclic presentation:
2^9|b^2=g,c^2=ghi,d^2=h,e^2=gi,f^2=gi,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, 8371434);