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 | #42024 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4276 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(3,7)(6,8)(9,10)(11,19)(12,16)(13,14)(15,18)(17,24)(20,23)(21,22)(25,26)(27,30)(28,33)(29,34)(31,35)(32,37)(36,38)(39,40);(1,2)(3,6)(4,5)(7,8)(9,11,15,12)(10,16,18,19)(13,20,17,22)(14,21,24,23)(26,27)(28,35)(32,34)(36,40);(2,5)(6,8)(9,12,15,11)(10,16,18,19)(13,20,17,22)(14,23,24,21)(25,27,30,26)(28,33,35,31)(29,34,37,32)(36,39,40,38);(1,2)(3,6)(4,5)(7,8)(9,10,15,18)(11,16,12,19)(13,14,17,24)(20,21,22,23)(25,28)(26,31)(27,33)(29,36)(30,35)(32,38)(34,39)(37,40);(1,2)(3,6)(4,5)(7,8)(9,13,15,17)(10,14,18,24)(11,20,12,22)(16,21,19,23)(25,29,30,37)(26,32,27,34)(28,36,35,40)(31,38,33,39);(1,3)(2,6)(4,7)(5,8)(9,14)(10,17)(11,21)(12,23)(13,18)(15,24)(16,22)(19,20)Polycyclic presentation:
2^9|b^2=gh,c^2=g,d^2=gh,e^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,e^c=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8032856);