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 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4246 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,15,19)(11,14,21,23)(12,18,22,16)(13,17,20,24)(25,26,32,36)(27,37,39,33)(28,31,34,29)(30,35,40,38);(1,2)(3,7)(4,5)(6,8)(9,11,15,21)(10,14,19,23)(12,13,22,20)(16,24,18,17)(25,27,32,39)(26,33,36,37)(28,30,34,40)(29,35,31,38);(2,5)(6,7)(9,12,15,22)(10,16,19,18)(11,20,21,13)(14,17,23,24)(25,28)(26,29)(27,30)(31,36)(32,34)(33,35)(37,38)(39,40);(2,5)(6,7)(9,13)(10,17)(11,12)(14,18)(15,20)(16,23)(19,24)(21,22)(25,29,32,31)(26,34,36,28)(27,38,39,35)(30,37,40,33);(1,2)(3,6)(4,5)(7,8)(9,12)(10,18)(11,13)(14,17)(15,22)(16,19)(20,21)(23,24)(25,30)(26,35)(27,34)(28,39)(29,37)(31,33)(32,40)(36,38);(1,3)(2,6)(4,8)(5,7)(9,14)(10,11)(12,17)(13,18)(15,23)(16,20)(19,21)(22,24)(25,31,32,29)(26,28,36,34)(27,35,39,38)(30,33,40,37)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=gh,d^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,f^b=fgi,e^c=egi,e^d=eghi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10228122);