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 | #46661 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4212 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)(11,17)(12,14)(13,16)(15,18)(19,23)(20,21)(22,24)(25,26,31,35)(27,29,33,40)(28,34,30,32)(36,39,37,38);(2,5)(6,7)(10,12)(13,21)(15,17)(19,24)(25,27)(26,29)(28,36)(30,37)(31,33)(32,38)(34,39)(35,40);(1,2)(3,6)(4,5)(7,8)(11,18)(15,17)(19,24)(22,23)(25,28,31,30)(26,32,35,34)(27,36,33,37)(29,38,40,39);(1,3,4,8)(2,6,5,7)(9,11,14,18)(10,15,12,17)(13,19,21,24)(16,22,20,23);(9,12,14,10)(11,17,18,15)(13,20,21,16)(19,23,24,22)(25,29)(26,33)(27,35)(28,38)(30,39)(31,40)(32,37)(34,36);(9,13)(10,16)(11,19)(12,20)(14,21)(15,22)(17,23)(18,24)(25,30,31,28)(26,34,35,32)(27,37,33,36)(29,39,40,38)Polycyclic presentation:
2^9|a^2=h,c^2=h,d^2=g,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eghi,f^b=fgi,d^c=dgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8663084);