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 | |
4427 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)(3,8)(4,6)(5,7)(9,10,15,19)(11,14,18,23)(12,22,21,16)(13,17,20,24)(26,32)(27,37)(28,36)(35,39);(1,3,5,4)(2,6,7,8)(9,11,15,18)(10,14,19,23)(12,13,21,20)(16,24,22,17)(25,26,31,32)(27,33,37,40)(28,30,36,34)(29,35,38,39);(9,12,15,21)(10,16,19,22)(11,20,18,13)(14,17,23,24)(25,27,31,37)(26,33,32,40)(28,38,36,29)(30,39,34,35);(2,7)(6,8)(9,11)(10,14)(12,20)(13,21)(15,18)(16,17)(19,23)(22,24)(25,28)(26,34)(27,38)(29,37)(30,32)(31,36)(33,35)(39,40);(1,4,5,3)(2,8,7,6)(9,13,15,20)(10,17,19,24)(11,21,18,12)(14,16,23,22)(25,29,31,38)(26,35,32,39)(27,28,37,36)(30,40,34,33);(1,2)(3,6)(4,8)(5,7)(9,14)(10,18)(11,19)(12,17)(13,16)(15,23)(20,22)(21,24)(25,30,31,34)(26,36,32,28)(27,39,37,35)(29,40,38,33)Polycyclic presentation:
2^9|a^2=i,b^2=gi,c^2=h,e^2=gi,f^2=hi,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, 8370635);