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 | #39402 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4444 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,6,5)(3,8,4,7)(12,20)(16,23)(19,24)(21,22)(26,31)(28,34)(32,38)(36,40);(1,3,6,4)(2,7,5,8)(9,10,14,18)(11,13,17,15)(12,16,20,23)(19,21,24,22)(27,35)(32,38)(36,40)(37,39);(9,11)(10,15)(12,19)(13,18)(14,17)(16,22)(20,24)(21,23)(25,26)(27,32)(28,33)(29,34)(30,31)(35,38)(36,39)(37,40);(2,5)(7,8)(9,11)(10,13)(12,19)(14,17)(15,18)(16,21)(20,24)(22,23)(25,27,30,35)(26,32,31,38)(28,36,34,40)(29,37,33,39);(1,4,6,3)(2,8,5,7)(9,12,14,20)(10,16,18,23)(11,19,17,24)(13,21,15,22)(25,28,30,34)(26,33,31,29)(27,36,35,40)(32,39,38,37);(1,5,6,2)(3,8,4,7)(9,13)(10,17)(11,18)(12,21)(14,15)(16,24)(19,23)(20,22)(25,29,30,33)(26,34,31,28)(27,37,35,39)(32,40,38,36)Polycyclic presentation:
2^9|a^2=g,b^2=gi,d^2=h,e^2=ghi,f^2=gh,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, 8352360);