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 | #2613 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4397 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)(11,15)(12,17)(13,18)(14,16)(19,21)(20,22)(23,24)(25,26)(27,33)(28,36)(29,32)(30,35)(31,38)(34,39)(37,40);(1,3,4,7)(2,5,6,8)(9,11,14,18)(10,15,16,13)(12,19,22,24)(17,21,20,23)(25,27)(26,30)(28,34)(29,35)(31,39)(32,33)(36,40)(37,38);(1,2)(3,5)(4,6)(7,8)(10,16)(11,18)(17,20)(19,24)(25,26,29,32)(27,30,35,33)(28,31,38,36)(34,39,37,40);(9,12)(10,17)(11,19)(13,23)(14,22)(15,21)(16,20)(18,24)(25,27)(26,30)(28,37)(29,35)(31,40)(32,33)(34,38)(36,39);(1,3)(2,5)(4,7)(6,8)(9,10)(11,13)(12,20)(14,16)(15,18)(17,22)(19,21)(23,24)(25,26)(27,33)(28,31)(29,32)(30,35)(34,40)(36,38)(37,39);(1,2)(3,5)(4,6)(7,8)(9,13)(10,11)(12,21)(14,15)(16,18)(17,24)(19,20)(22,23)(25,28)(26,31)(27,34)(29,38)(30,39)(32,36)(33,40)(35,37)Polycyclic presentation:
2^9|a^2=ghi,b^2=gh,c^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^c=egi,e^d=egi,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8973061);