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 | #54792 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4151 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,14,18)(11,15,12,16)(13,17,21,24)(19,22,20,23)(25,26,32,36)(27,37,40,33)(28,29,30,34)(31,39,38,35);(1,2,5,8)(3,4,7,6)(9,10)(11,15)(12,16)(13,17)(14,18)(19,22)(20,23)(21,24)(25,27)(26,33)(28,38)(29,39)(30,31)(32,40)(34,35)(36,37);(1,3)(2,6)(4,8)(5,7)(9,11,14,12)(10,15,18,16)(13,19,21,20)(17,22,24,23)(25,28,32,30)(26,34,36,29)(27,38,40,31)(33,35,37,39);(9,10)(11,15)(12,16)(13,17)(14,18)(19,22)(20,23)(21,24)(25,29,32,34)(26,28,36,30)(27,39,40,35)(31,33,38,37);(1,3)(2,4)(5,7)(6,8)(9,12,14,11)(10,16,18,15)(13,19,21,20)(17,22,24,23)(25,30,32,28)(26,34,36,29)(27,38,40,31)(33,39,37,35);(1,4)(2,7)(3,8)(5,6)(9,13)(10,17)(11,19)(12,20)(14,21)(15,22)(16,23)(18,24)(25,31,32,38)(26,35,36,39)(27,30,40,28)(29,33,34,37)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=g,d^2=h,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9583808);