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 | #2416 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4375 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)(4,6)(5,8)(9,10,14,18)(11,15,16,12)(13,17,22,21)(19,23,24,20)(25,26)(27,33)(28,30)(29,32)(31,35)(34,39)(36,37)(38,40);(1,2,4,6)(3,5,8,7)(9,10)(11,15)(12,16)(13,17)(14,18)(19,23)(20,24)(21,22)(26,28)(31,33)(32,36)(38,39);(1,2)(3,5)(4,6)(7,8)(9,11,14,16)(10,15,18,12)(13,19,22,24)(17,23,21,20)(25,27)(26,31)(28,33)(29,34)(30,35)(32,38)(36,39)(37,40);(1,3)(2,5)(4,8)(6,7)(9,10)(11,15)(12,16)(13,21)(14,18)(17,22)(19,20)(23,24)(25,28,30,26)(27,33,35,31)(29,32,37,36)(34,38,40,39);(9,12,14,15)(10,16,18,11)(13,20,22,23)(17,24,21,19);(9,13,14,22)(10,17,18,21)(11,19,16,24)(12,20,15,23)(25,29,30,37)(26,32,28,36)(27,34,35,40)(31,38,33,39)Polycyclic presentation:
2^9|a^2=ghi,b^2=i,c^2=ghi,d^2=h,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, 8374426);