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 | C22×#41 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4301 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,14,16)(11,15,12,17)(13,18,21,23)(19,22,20,24)(25,26)(27,38)(28,33)(29,35)(30,36)(31,37)(32,40)(34,39);(1,2,5,6)(3,7,8,4)(9,11)(10,15)(12,14)(13,19)(16,17)(18,22)(20,21)(23,24)(25,27)(26,32)(28,29)(30,33)(31,40)(34,35)(36,39)(37,38);(1,2)(3,7)(4,8)(5,6)(10,16)(11,12)(13,21)(22,24)(25,28,31,39)(26,33,37,34)(27,29,40,36)(30,38,35,32);(2,6)(3,8)(9,12,14,11)(10,17,16,15)(13,19,21,20)(18,22,23,24)(25,28)(26,34)(27,36)(29,40)(30,32)(31,39)(33,37)(35,38);(1,3)(2,4)(5,8)(6,7)(9,13,14,21)(10,18,16,23)(11,19,12,20)(15,22,17,24)(25,29,31,36)(26,35,37,30)(27,39,40,28)(32,33,38,34);(1,4)(2,3)(5,7)(6,8)(9,10)(11,15)(12,17)(13,18)(14,16)(19,22)(20,24)(21,23)(25,30,31,35)(26,36,37,29)(27,33,40,34)(28,32,39,38)Polycyclic presentation:
2^9|a^2=h,b^2=hi,c^2=g,d^2=h,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,d^c=dgi,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, 8968646);