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 | #34732 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4445 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)(4,6)(5,8)(9,10,16,20)(11,21,23,17)(12,19,14,18)(13,22,24,15)(25,26,30,33)(27,31,34,38)(28,29,32,37)(35,36,39,40);(1,2,4,6)(3,5,8,7)(9,11,16,23)(10,17,20,21)(12,22,14,15)(13,18,24,19)(27,34)(31,38)(35,39)(36,40);(1,2)(3,5)(4,6)(7,8)(9,12,16,14)(10,18,20,19)(11,22,23,15)(13,17,24,21)(25,27,30,34)(26,31,33,38)(28,35,32,39)(29,36,37,40);(1,3)(2,5)(4,8)(6,7)(9,13,16,24)(10,15,20,22)(11,19,23,18)(12,17,14,21)(25,28)(26,29)(27,35)(30,32)(31,36)(33,37)(34,39)(38,40);(9,14,16,12)(10,19,20,18)(11,15,23,22)(13,21,24,17);(9,15)(10,13)(11,12)(14,23)(16,22)(17,18)(19,21)(20,24)(25,29)(26,32)(27,36)(28,33)(30,37)(31,39)(34,40)(35,38)Polycyclic presentation:
2^9|a^2=gi,b^2=g,c^2=gi,d^2=h,e^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,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, 8332833);