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 | #2541 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4379 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)(3,4)(5,7)(6,8)(12,20)(17,21)(18,24)(22,23)(25,26,32,30)(27,37,39,33)(28,35,29,34)(31,36,38,40);(1,3)(2,6)(4,7)(5,8)(9,10,14,13)(11,16,19,15)(12,17,20,21)(18,23,24,22)(25,27)(26,33)(28,38)(29,31)(30,37)(32,39)(34,40)(35,36);(9,11)(10,15)(12,18)(13,16)(14,19)(17,22)(20,24)(21,23)(25,28,32,29)(26,34,30,35)(27,38,39,31)(33,40,37,36);(2,7)(4,6)(9,11)(10,16)(12,18)(13,15)(14,19)(17,23)(20,24)(21,22)(25,29,32,28)(26,35,30,34)(27,38,39,31)(33,40,37,36);(1,3)(2,6)(4,7)(5,8)(9,12,14,20)(10,17,13,21)(11,18,19,24)(15,22,16,23)(25,30,32,26)(27,37,39,33)(28,35,29,34)(31,40,38,36);(1,4,5,6)(2,3,7,8)(9,13,14,10)(11,16,19,15)(12,21,20,17)(18,23,24,22)(25,31,32,38)(26,36,30,40)(27,29,39,28)(33,35,37,34)Polycyclic presentation:
2^9|a^2=h,b^2=i,c^2=h,d^2=h,e^2=hi,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, 8368793);