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 | #52297 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4449 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2)(3,7)(4,5)(6,8)(9,10)(11,14)(12,15)(13,16)(17,20)(18,22)(19,23)(21,24)(25,26,30,34)(27,29,33,31)(28,32,36,39)(35,37,40,38);(2,4)(6,7)(10,11)(15,17)(16,18)(23,24)(28,36)(32,39)(35,40)(37,38);(1,3,5,8)(2,6,4,7)(9,10,14,11)(12,15,20,17)(13,16,22,18)(19,23,21,24)(25,27)(26,31)(28,35)(29,34)(30,33)(32,38)(36,40)(37,39);(1,4,5,2)(3,7,8,6)(9,11,14,10)(12,17,20,15)(13,18,22,16)(19,24,21,23)(25,28)(26,32)(27,35)(29,37)(30,36)(31,38)(33,40)(34,39);(9,12,14,20)(10,15,11,17)(13,21,22,19)(16,24,18,23)(25,27)(26,29)(28,35)(30,33)(31,34)(32,37)(36,40)(38,39);(9,13,14,22)(10,16,11,18)(12,19,20,21)(15,23,17,24)(25,29)(26,33)(27,34)(28,37)(30,31)(32,40)(35,39)(36,38)Polycyclic presentation:
2^9|a^2=gh,c^2=g,d^2=g,e^2=i,f^2=i,b^a=bg,c^a=ch,d^a=dg,c^b=ci,d^b=dh,f^e=fghi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8003300);