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 | #52981 | 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)(3,6)(4,8)(5,7)(9,10)(11,13)(12,20)(14,15)(16,18)(17,22)(19,23)(21,24)(25,26,30,34)(27,31,32,28)(29,37,38,33)(35,36,40,39);(2,3)(7,8)(10,15)(11,18)(17,20)(19,24)(25,27)(26,31)(28,34)(29,35)(30,32)(33,39)(36,37)(38,40);(2,3)(7,8)(9,11)(10,16)(12,19)(13,15)(14,18)(17,23)(20,21)(22,24)(25,28,30,31)(26,32,34,27)(29,36,38,39)(33,40,37,35);(1,3,6,2)(4,8,5,7)(9,11)(10,16)(12,19)(13,15)(14,18)(17,23)(20,21)(22,24)(25,28)(26,27)(29,36)(30,31)(32,34)(33,35)(37,40)(38,39);(1,4,6,5)(2,7,3,8)(9,12,14,22)(10,17,15,20)(11,19,18,24)(13,21,16,23)(25,29,30,38)(26,33,34,37)(27,35,32,40)(28,36,31,39);(1,5,6,4)(2,8,3,7)(9,13,14,16)(10,11,15,18)(12,21,22,23)(17,19,20,24)(25,27)(26,28)(29,35)(30,32)(31,34)(33,36)(37,39)(38,40)Polycyclic presentation:
2^9|a^2=gh,c^2=gh,d^2=hi,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8014475);