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 | #52292 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4485 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)(9,10,14,15)(11,12,17,16)(13,18,23,24)(19,20,21,22)(25,26,31,37)(27,32,39,40)(28,34,38,33)(29,30,36,35);(3,7)(6,8)(10,15)(11,17)(18,24)(19,21)(25,27)(26,32)(28,35)(29,34)(30,38)(31,39)(33,36)(37,40);(1,2)(3,6)(4,5)(7,8)(25,28)(26,33)(27,30)(29,32)(31,38)(34,37)(35,39)(36,40);(1,3)(2,6)(4,7)(5,8)(9,11,14,17)(10,16,15,12)(13,21,23,19)(18,20,24,22)(25,28)(26,34)(27,35)(29,32)(30,39)(31,38)(33,37)(36,40);(1,2)(3,6)(4,5)(7,8)(9,12,14,16)(10,17,15,11)(13,22,23,20)(18,19,24,21)(25,29)(26,35)(27,34)(28,32)(30,37)(31,36)(33,39)(38,40);(9,13,14,23)(10,18,15,24)(11,19,17,21)(12,20,16,22)(25,30,31,35)(26,36,37,29)(27,38,39,28)(32,33,40,34)Polycyclic presentation:
2^9|a^2=gi,d^2=g,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,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, 8606564);