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 | #50873 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4312 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,14,17)(11,16,20,15)(12,21,22,13)(18,19,24,23)(25,26,31,32)(27,29,28,33)(30,34,38,39)(35,37,36,40);(1,2)(3,6)(4,7)(5,8)(9,11)(10,15)(12,18)(13,19)(14,20)(16,17)(21,23)(22,24)(26,32)(27,28)(34,39)(35,36);(1,3)(2,5)(4,8)(6,7)(25,27,31,28)(26,33,32,29)(30,35,38,36)(34,40,39,37);(1,3)(2,6)(4,8)(5,7)(9,12,14,22)(10,13,17,21)(11,18,20,24)(15,19,16,23)(25,28,31,27)(26,29,32,33)(30,36,38,35)(34,37,39,40);(1,3)(2,6)(4,8)(5,7)(9,11)(10,16)(12,18)(13,23)(14,20)(15,17)(19,21)(22,24)(25,29)(26,27)(28,32)(30,37)(31,33)(34,35)(36,39)(38,40);(1,2)(3,5)(4,7)(6,8)(9,13)(10,12)(11,19)(14,21)(15,18)(16,24)(17,22)(20,23)(25,30,31,38)(26,34,32,39)(27,35,28,36)(29,37,33,40)Polycyclic presentation:
2^9|a^2=g,c^2=h,d^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^d=fgi,f^e=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8566610);