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 | #35906 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4573 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,4)(6,8)(9,10,15,19)(11,16,20,22)(12,17,21,24)(13,14,18,23)(25,26)(27,34)(28,36)(29,32)(30,33)(31,37)(35,39)(38,40);(3,7)(6,8)(9,11,15,20)(10,16,19,22)(12,14,21,23)(13,17,18,24)(25,27,30,32)(26,29,33,34)(28,35,37,40)(31,39,36,38);(1,2,5,4)(3,6,7,8)(11,20)(12,21)(16,22)(17,24)(28,37)(31,36)(35,40)(38,39);(1,3,5,7)(2,6,4,8)(9,12)(10,17)(11,14)(13,22)(15,21)(16,18)(19,24)(20,23)(25,28,30,37)(26,31,33,36)(27,35,32,40)(29,39,34,38);(1,4,5,2)(3,8,7,6)(9,13)(10,14)(11,17)(12,22)(15,18)(16,21)(19,23)(20,24)(25,26)(27,29)(28,31)(30,33)(32,34)(35,39)(36,37)(38,40);(9,14)(10,18)(11,21)(12,20)(13,19)(15,23)(16,24)(17,22)(25,29)(26,32)(27,33)(28,38)(30,34)(31,35)(36,40)(37,39)Polycyclic presentation:
2^9|a^2=i,b^2=gi,c^2=h,d^2=gh,e^2=h,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, 8619909);