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 | #50886 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4237 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)(3,8)(4,6)(5,7)(9,10,15,19)(11,16,20,24)(12,14,18,23)(13,22,21,17)(25,26,30,31)(27,29,34,32)(28,36,38,33)(35,39,40,37);(1,3,6,5)(2,7,4,8)(11,20)(13,21)(16,24)(17,22)(26,31)(27,34)(33,36)(35,40);(9,11,15,20)(10,16,19,24)(12,21,18,13)(14,17,23,22)(25,27,30,34)(26,32,31,29)(28,35,38,40)(33,39,36,37);(1,2,6,4)(3,7,5,8)(9,12)(10,14)(11,21)(13,20)(15,18)(16,17)(19,23)(22,24)(25,26)(27,32)(28,33)(29,34)(30,31)(35,39)(36,38)(37,40);(1,4,6,2)(3,7,5,8)(9,13,15,21)(10,17,19,22)(11,12,20,18)(14,16,23,24)(25,28)(26,33)(27,35)(29,37)(30,38)(31,36)(32,39)(34,40);(1,5,6,3)(2,7,4,8)(9,14)(10,18)(11,17)(12,19)(13,16)(15,23)(20,22)(21,24)(25,29,30,32)(26,34,31,27)(28,37,38,39)(33,40,36,35)Polycyclic presentation:
2^9|a^2=i,b^2=gh,c^2=i,d^2=gh,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,e^b=egi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8400103);