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 | #40723 | 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:
(1,2)(3,7)(4,5)(6,8)(9,10)(11,19)(12,23)(13,24)(14,18)(15,22)(16,21)(17,20)(26,32)(27,28)(33,39)(34,36);(2,5)(6,7)(9,11)(10,15)(12,20)(13,21)(14,22)(16,17)(18,19)(23,24)(25,26,31,32)(27,30,28,35)(29,33,38,39)(34,37,36,40);(1,2)(3,6)(4,5)(7,8)(25,27,31,28)(26,30,32,35)(29,34,38,36)(33,37,39,40);(1,3,4,8)(2,6,5,7)(9,12,14,21)(10,16,18,23)(11,20,22,13)(15,17,19,24)(25,28,31,27)(26,30,32,35)(29,36,38,34)(33,37,39,40);(9,13,14,20)(10,17,18,24)(11,21,22,12)(15,16,19,23)(25,29,31,38)(26,33,32,39)(27,34,28,36)(30,37,35,40);(9,10)(11,15)(12,23)(13,17)(14,18)(16,21)(19,22)(20,24)(25,30)(26,28)(27,32)(29,37)(31,35)(33,36)(34,39)(38,40)Polycyclic presentation:
2^9|b^2=h,c^2=h,d^2=g,e^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8345245);