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 | #38540 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4568 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,5)(6,8)(9,10)(11,18)(12,20)(13,15)(14,17)(16,21)(19,23)(22,24)(25,26)(27,31)(28,34)(29,33)(30,32)(35,38)(36,40)(37,39);(3,7)(6,8)(9,11,14,13)(10,15,17,18)(12,19,21,22)(16,23,20,24)(25,27,30,34)(26,31,32,28)(29,35,39,40)(33,38,37,36);(1,2)(3,6)(4,5)(7,8)(12,21)(16,20)(19,22)(23,24)(26,32)(27,34)(33,37)(35,40);(1,3)(2,6)(4,7)(5,8)(9,12)(10,16)(11,19)(13,22)(14,21)(15,23)(17,20)(18,24)(25,26)(27,28)(29,37)(30,32)(31,34)(33,39)(35,38)(36,40);(1,2)(3,6)(4,5)(7,8)(9,13,14,11)(10,15,17,18)(12,22,21,19)(16,23,20,24)(25,28)(26,27)(29,38)(30,31)(32,34)(33,40)(35,37)(36,39);(9,10)(11,15)(12,20)(13,18)(14,17)(16,21)(19,24)(22,23)(25,29)(26,33)(27,35)(28,36)(30,39)(31,38)(32,37)(34,40)Polycyclic presentation:
2^9|b^2=gi,e^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,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, 8616219);