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 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4637 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)(3,7)(9,10)(11,12)(13,17)(14,15)(16,18)(19,20)(21,22)(23,24);(1,2,4,5)(3,6,7,8)(9,10)(11,18)(12,16)(13,17)(14,15)(19,22)(20,21)(23,24)(25,26)(27,35)(28,32)(29,33)(30,34)(31,36)(37,39)(38,40);(1,2)(3,6)(4,5)(7,8)(10,15)(11,16)(17,24)(19,21)(25,27,30,36)(26,31,34,35)(28,37,38,29)(32,33,40,39);(1,2)(3,6)(4,5)(7,8)(9,11,14,16)(10,12,15,18)(13,21,23,19)(17,22,24,20)(25,28,30,38)(26,32,34,40)(27,29,36,37)(31,39,35,33);(1,3)(2,6)(4,7)(5,8)(9,12)(10,16)(11,15)(13,22)(14,18)(17,19)(20,23)(21,24)(25,26)(27,31)(28,32)(29,39)(30,34)(33,37)(35,36)(38,40);(9,13,14,23)(10,17,15,24)(11,19,16,21)(12,20,18,22)(25,29)(26,33)(27,28)(30,37)(31,40)(32,35)(34,39)(36,38)Polycyclic presentation:
2^9|b^2=g,c^2=hi,d^2=gi,f^2=gh,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, 8626105);