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 |
| prime radical | #41341 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4269 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)(4,5)(6,8)(9,10,15,16)(11,17,12,13)(14,18,22,23)(19,24,20,21)(26,31)(28,32)(29,37)(35,38);(2,5)(6,7)(10,16)(13,17)(18,23)(21,24)(26,31)(28,32)(34,39)(36,40);(1,2)(3,6)(4,5)(7,8)(9,11,15,12)(10,17,16,13)(14,19,22,20)(18,24,23,21)(25,26)(27,32)(28,33)(29,34)(30,31)(35,40)(36,38)(37,39);(1,3)(2,6)(4,8)(5,7)(9,12,15,11)(10,17,16,13)(14,19,22,20)(18,21,23,24)(25,27,30,33)(26,32,31,28)(29,38,37,35)(34,36,39,40);(9,13,15,17)(10,12,16,11)(14,21,22,24)(18,20,23,19)(25,28,30,32)(26,33,31,27)(29,36,37,40)(34,38,39,35);(9,14)(10,18)(11,19)(12,20)(13,21)(15,22)(16,23)(17,24)(25,29,30,37)(26,34,31,39)(27,35,33,38)(28,36,32,40)Polycyclic presentation:
2^9|a^2=ghi,c^2=ghi,d^2=gi,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=egi,e^d=eghi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10352182);