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 | |
4449 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)(3,4)(5,6)(7,8)(9,10,15,18)(11,16,13,12)(14,17,23,22)(19,24,21,20)(26,31)(27,34)(28,36)(37,39);(2,6)(3,7)(9,11,15,13)(10,12,18,16)(14,19,23,21)(17,20,22,24)(25,26,30,31)(27,29,34,32)(28,33,36,38)(35,37,40,39);(1,3,5,7)(2,4,6,8)(9,12,15,16)(10,11,18,13)(14,20,23,24)(17,19,22,21)(25,27,30,34)(26,32,31,29)(28,35,36,40)(33,39,38,37);(1,4)(2,7)(3,6)(5,8)(9,10)(11,12)(13,16)(14,22)(15,18)(17,23)(19,24)(20,21)(25,26)(27,32)(28,33)(29,34)(30,31)(35,39)(36,38)(37,40);(9,13,15,11)(10,16,18,12)(14,21,23,19)(17,24,22,20)(25,28,30,36)(26,33,31,38)(27,35,34,40)(29,37,32,39);(9,14,15,23)(10,17,18,22)(11,19,13,21)(12,20,16,24)(25,29,30,32)(26,34,31,27)(28,37,36,39)(33,40,38,35)Polycyclic presentation:
2^9|a^2=gi,b^2=ghi,c^2=g,e^2=ghi,f^2=ghi,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, 8370880);