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 | #36396 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4270 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,5)(3,7)(9,10,15,20)(11,17,12,16)(13,21,23,18)(14,19,24,22)(26,28)(31,34)(32,36)(38,40);(1,2,4,5)(3,6,7,8)(9,11,15,12)(10,16,20,17)(13,22,23,19)(14,21,24,18)(27,33)(31,34)(35,39)(38,40);(1,2)(3,6)(4,5)(7,8)(25,26,30,28)(27,31,33,34)(29,32,37,36)(35,38,39,40);(1,2)(3,6)(4,5)(7,8)(9,12,15,11)(10,17,20,16)(13,22,23,19)(14,21,24,18)(25,27)(26,31)(28,34)(29,35)(30,33)(32,38)(36,40)(37,39);(1,3,4,7)(2,6,5,8)(9,13,15,23)(10,18,20,21)(11,19,12,22)(14,16,24,17)(25,28,30,26)(27,34,33,31)(29,36,37,32)(35,40,39,38);(9,14)(10,19)(11,21)(12,18)(13,16)(15,24)(17,23)(20,22)(25,29,30,37)(26,32,28,36)(27,35,33,39)(31,38,34,40)Polycyclic presentation:
2^9|a^2=gi,b^2=g,c^2=hi,d^2=gi,e^2=ghi,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,e^b=egi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8416132);