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 | #39953 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4525 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,4,6)(3,5,7,8)(9,10)(11,19)(12,18)(13,16)(14,20)(15,17)(21,22)(23,24)(26,30)(27,33)(32,36)(35,37);(1,2)(3,5)(4,6)(7,8)(9,11,15,13)(10,16,17,19)(12,21,20,23)(14,22,18,24)(25,26)(27,31)(28,36)(29,30)(32,38)(33,34)(35,40)(37,39);(10,17)(12,20)(16,19)(21,23)(25,26,29,30)(27,34,33,31)(28,32,38,36)(35,40,37,39);(9,12,15,20)(10,18,17,14)(11,21,13,23)(16,24,19,22)(25,26)(27,31)(28,32)(29,30)(33,34)(35,39)(36,38)(37,40);(1,3,4,7)(2,5,6,8)(9,13,15,11)(10,19,17,16)(12,23,20,21)(14,24,18,22)(25,27,29,33)(26,31,30,34)(28,37,38,35)(32,40,36,39);(9,14,15,18)(10,20,17,12)(11,22,13,24)(16,23,19,21)(25,28)(26,32)(27,35)(29,38)(30,36)(31,39)(33,37)(34,40)Polycyclic presentation:
2^9|a^2=gh,b^2=hi,c^2=i,d^2=hi,e^2=g,f^2=hi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,f^b=fi,d^c=dh,e^c=ei,f^e=fi101 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7895066);