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 | |
4446 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,7)(3,4)(9,10,16,15)(11,19,14,17)(12,18,13,22)(20,24,21,23)(25,26)(27,32)(28,33)(29,39)(30,35)(31,36)(34,38)(37,40);(1,2)(3,8)(4,5)(6,7)(9,11,16,14)(10,17,15,19)(12,20,13,21)(18,23,22,24)(25,26)(27,37)(28,33)(29,34)(30,35)(31,36)(32,40)(38,39);(1,3,6,4)(2,8,7,5)(9,12,16,13)(10,18,15,22)(11,20,14,21)(17,23,19,24)(25,27,31,40)(26,32,36,37)(28,38,35,29)(30,39,33,34);(1,4,6,3)(2,8,7,5)(9,13,16,12)(10,18,15,22)(11,21,14,20)(17,23,19,24)(25,28)(26,33)(27,38)(29,40)(30,36)(31,35)(32,34)(37,39);(9,14,16,11)(10,19,15,17)(12,21,13,20)(18,24,22,23)(25,29,31,38)(26,34,36,39)(27,28,40,35)(30,37,33,32);(1,5,6,8)(2,4,7,3)(9,15,16,10)(11,19,14,17)(12,22,13,18)(20,24,21,23)(25,30,31,33)(26,35,36,28)(27,39,40,34)(29,37,38,32)Polycyclic presentation:
2^9|a^2=gh,b^2=gh,c^2=gi,d^2=g,e^2=ghi,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, 8371590);