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 | #51893 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4501 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)(10,16)(11,19)(12,21)(18,23)(25,26,30,32)(27,33,34,28)(29,31,38,40)(35,37,39,36);(1,3)(2,5)(4,8)(6,7)(9,10,15,16)(11,14,19,24)(12,22,21,17)(13,18,20,23)(27,34)(28,33)(35,39)(36,37);(9,11)(10,14)(12,20)(13,21)(15,19)(16,24)(17,18)(22,23)(25,27)(26,28)(29,35)(30,34)(31,36)(32,33)(37,40)(38,39);(1,2,4,6)(3,5,8,7)(9,12,15,21)(10,17,16,22)(11,20,19,13)(14,18,24,23)(25,28)(26,27)(29,37)(30,33)(31,39)(32,34)(35,40)(36,38);(1,3)(2,5)(4,8)(6,7)(9,13,15,20)(10,18,16,23)(11,21,19,12)(14,17,24,22);(1,2)(3,5)(4,6)(7,8)(9,14,15,24)(10,19,16,11)(12,23,21,18)(13,17,20,22)(25,29)(26,31)(27,35)(28,36)(30,38)(32,40)(33,37)(34,39)Polycyclic presentation:
2^9|a^2=i,b^2=hi,d^2=g,e^2=hi,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,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, 8370845);