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 | #36713 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4357 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)(11,13)(12,17)(14,23)(15,16)(18,20)(19,22)(21,24)(26,32)(28,36)(33,38)(34,37);(2,5)(6,7)(9,11)(10,13)(12,20)(14,21)(15,19)(16,22)(17,18)(23,24)(25,26)(27,33)(28,30)(29,38)(31,32)(34,35)(36,40)(37,39);(1,2)(3,6)(4,5)(7,8)(10,16)(11,19)(12,21)(18,23)(25,27,31,29)(26,33,32,38)(28,34,36,37)(30,35,40,39);(9,12,15,21)(10,17,16,24)(11,20,19,14)(13,18,22,23)(25,28)(26,30)(27,34)(29,37)(31,36)(32,40)(33,35)(38,39);(1,2)(3,7)(4,5)(6,8)(9,13)(10,11)(12,18)(14,24)(15,22)(16,19)(17,20)(21,23)(25,29,31,27)(26,33,32,38)(28,37,36,34)(30,35,40,39);(1,3)(2,6)(4,8)(5,7)(9,14,15,20)(10,18,16,23)(11,21,19,12)(13,17,22,24)(25,30)(26,28)(27,35)(29,39)(31,40)(32,36)(33,34)(37,38)Polycyclic presentation:
2^9|c^2=g,d^2=h,e^2=g,f^2=h,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, 9688244);