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 | #47939 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4519 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,5)(6,8)(9,10,13,15)(11,14,18,17)(12,20,21,16)(19,23,24,22)(26,31)(28,38)(32,33)(35,36);(3,7)(6,8)(9,11)(10,14)(12,19)(13,18)(15,17)(16,22)(20,23)(21,24)(25,26)(27,32)(28,39)(29,35)(30,31)(33,37)(34,38)(36,40);(1,2)(3,6)(4,5)(7,8)(10,15)(11,18)(16,20)(19,24)(25,27,30,37)(26,32,31,33)(28,36,38,35)(29,39,40,34);(9,10)(11,17)(12,20)(13,15)(14,18)(16,21)(19,22)(23,24)(25,27)(26,33)(28,35)(29,34)(30,37)(31,32)(36,38)(39,40);(1,3,4,7)(2,6,5,8)(9,12,13,21)(10,16,15,20)(11,19,18,24)(14,22,17,23)(25,28)(26,34)(27,36)(29,32)(30,38)(31,39)(33,40)(35,37);(1,2)(3,6)(4,5)(7,8)(9,11)(10,17)(12,19)(13,18)(14,15)(16,23)(20,22)(21,24)(25,29,30,40)(26,35,31,36)(27,34,37,39)(28,32,38,33)Polycyclic presentation:
2^9|a^2=i,c^2=g,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=di,e^a=egi,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dgi,f^c=fgi,e^d=egi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8841087);