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 | |
4489 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,15,17)(11,20,22,16)(12,14,23,18)(13,19,24,21)(25,26)(27,33)(28,39)(29,31)(30,34)(32,35)(36,38)(37,40);(2,6)(3,8)(9,11,15,22)(10,16,17,20)(12,19,23,21)(13,18,24,14)(25,27,32,30)(26,33,35,34)(28,37,29,36)(31,38,39,40);(1,2)(3,7)(4,8)(5,6)(10,17)(13,24)(14,18)(16,20)(25,28,32,29)(26,31,35,39)(27,36,30,37)(33,40,34,38);(1,3,5,8)(2,7,6,4)(9,12)(10,18)(11,21)(13,16)(14,17)(15,23)(19,22)(20,24)(25,29,32,28)(26,31,35,39)(27,37,30,36)(33,40,34,38);(1,3)(2,4)(5,8)(6,7)(9,13)(10,19)(11,18)(12,20)(14,22)(15,24)(16,23)(17,21)(25,30,32,27)(26,34,35,33)(28,36,29,37)(31,40,39,38);(1,4)(2,3)(5,7)(6,8)(9,14,15,18)(10,12,17,23)(11,13,22,24)(16,19,20,21)(25,31)(26,28)(27,38)(29,35)(30,40)(32,39)(33,37)(34,36)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=gh,d^2=i,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,e^c=egi,f^c=fgi,e^d=egi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8948217);