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 | |
4562 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,8)(3,6,4,7)(9,10)(11,18)(12,20)(13,15)(14,17)(16,22)(19,23)(21,24)(25,26)(27,32)(28,33)(29,34)(30,31)(35,37)(36,40)(38,39);(1,2)(3,6)(4,7)(5,8)(26,31)(27,33)(34,39)(35,40);(9,11,14,13)(10,15,17,18)(12,19,22,21)(16,23,20,24)(25,27,30,33)(26,32,31,28)(29,35,38,40)(34,37,39,36);(9,12)(10,16)(11,19)(13,21)(14,22)(15,23)(17,20)(18,24);(1,3,5,4)(2,6,8,7)(9,13,14,11)(10,15,17,18)(12,21,22,19)(16,23,20,24)(25,28,30,32)(26,33,31,27)(29,37,38,36)(34,35,39,40);(1,4,5,3)(2,7,8,6)(9,10)(11,15)(12,16)(13,18)(14,17)(19,23)(20,22)(21,24)(25,29,30,38)(26,34,31,39)(27,35,33,40)(28,36,32,37)Polycyclic presentation:
2^9|a^2=gi,c^2=hi,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dh,c^b=ci,f^e=fhi104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7562126);