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 | #40537 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4526 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,14,16)(11,19,20,15)(12,21,23,17)(13,18,22,24)(25,26)(27,36)(28,30)(29,39)(31,34)(32,35)(33,38)(37,40);(1,2)(3,6)(4,5)(7,8)(9,11,14,20)(10,15,16,19)(12,22,23,13)(17,24,21,18)(25,27)(26,33)(28,37)(29,34)(30,31)(32,38)(35,36)(39,40);(2,5)(6,8)(10,16)(15,19)(17,21)(18,24)(25,28,32,29)(26,30,35,39)(27,37,38,34)(31,36,40,33);(3,7)(6,8)(9,12,14,23)(10,17,16,21)(11,13,20,22)(15,18,19,24)(25,29,32,28)(26,30,35,39)(27,34,38,37)(31,36,40,33);(1,2)(3,6)(4,5)(7,8)(9,10)(11,15)(12,17)(13,18)(14,16)(19,20)(21,23)(22,24)(25,30)(26,28)(27,31)(29,35)(32,39)(33,37)(34,36)(38,40);(1,3)(2,6)(4,7)(5,8)(9,13)(10,18)(11,12)(14,22)(15,17)(16,24)(19,21)(20,23)(25,31)(26,34)(27,30)(28,36)(29,33)(32,40)(35,37)(38,39)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=gh,d^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=egi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8416673);