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 | #35212 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4246 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,4,6)(3,5,7,8)(9,10,15,12)(11,17,13,16)(14,18,22,20)(19,24,21,23)(29,37)(33,38)(34,39)(36,40);(1,2)(3,5)(4,6)(7,8)(9,11,15,13)(10,16,12,17)(14,19,22,21)(18,23,20,24)(25,26)(27,28)(29,33)(30,31)(32,35)(34,36)(37,38)(39,40);(1,3,4,7)(2,5,6,8)(10,12)(16,17)(18,20)(23,24)(26,31)(28,35)(33,38)(36,40);(9,12,15,10)(11,17,13,16)(14,20,22,18)(19,24,21,23)(25,27,30,32)(26,28,31,35)(29,34,37,39)(33,36,38,40);(9,13,15,11)(10,17,12,16)(14,21,22,19)(18,24,20,23)(25,28)(26,32)(27,31)(29,36)(30,35)(33,39)(34,38)(37,40);(9,14,15,22)(10,18,12,20)(11,19,13,21)(16,23,17,24)(25,29,30,37)(26,33,31,38)(27,34,32,39)(28,36,35,40)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=gh,d^2=hi,e^2=h,f^2=hi,b^a=bg,c^a=ch,e^a=eh,f^a=fi,c^b=ci,e^b=ei,d^c=dh,e^c=ei98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7628919);