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 | #51687 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4350 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,15,19)(11,20,23,16)(12,21,22,17)(13,14,18,24)(25,26)(27,29)(28,36)(30,34)(31,33)(32,38)(35,39)(37,40);(1,2,4,6)(3,5,8,7)(9,11,15,23)(10,16,19,20)(12,14,22,24)(13,17,18,21)(25,27)(26,31)(28,35)(29,34)(30,33)(32,39)(36,37)(38,40);(1,2)(3,5)(4,6)(7,8)(25,28,30,38)(26,32,34,36)(27,35,33,40)(29,37,31,39);(9,12,15,22)(10,17,19,21)(11,14,23,24)(13,16,18,20)(25,28)(26,32)(27,35)(29,37)(30,38)(31,39)(33,40)(34,36);(1,2)(3,7)(4,6)(5,8)(9,13)(10,14)(11,21)(12,16)(15,18)(17,23)(19,24)(20,22)(25,27)(26,29)(28,35)(30,33)(31,34)(32,37)(36,39)(38,40);(1,3)(2,5)(4,8)(6,7)(9,14)(10,18)(11,22)(12,23)(13,19)(15,24)(16,21)(17,20)(25,29)(26,33)(27,34)(28,37)(30,31)(32,40)(35,36)(38,39)Polycyclic presentation:
2^9|a^2=gh,b^2=ghi,c^2=h,d^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9694684);