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×D4 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4290 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,8)(3,6,4,7)(9,10,15,17)(11,16,12,14)(13,21,22,18)(19,24,20,23)(25,26)(27,33)(28,30)(29,39)(31,34)(32,35)(36,40)(37,38);(1,2)(3,6)(4,7)(5,8)(9,11,15,12)(10,16,17,14)(13,19,22,20)(18,23,21,24)(25,27,32,29)(26,33,35,39)(28,38,31,36)(30,37,34,40);(10,17)(11,12)(13,22)(23,24)(25,28,32,31)(26,30,35,34)(27,36,29,38)(33,40,39,37);(1,3,5,4)(2,6,8,7)(9,12,15,11)(10,16,17,14)(13,20,22,19)(18,23,21,24)(25,29,32,27)(26,33,35,39)(28,38,31,36)(30,40,34,37);(1,4,5,3)(2,7,8,6)(9,13,15,22)(10,18,17,21)(11,19,12,20)(14,24,16,23)(25,30)(26,28)(27,37)(29,40)(31,35)(32,34)(33,38)(36,39);(9,14,15,16)(10,11,17,12)(13,23,22,24)(18,20,21,19)(25,31,32,28)(26,34,35,30)(27,38,29,36)(33,37,39,40)Polycyclic presentation:
2^9|a^2=gh,b^2=i,c^2=hi,d^2=gi,e^2=gh,f^2=i,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,f^b=fi,d^c=dh,e^c=ei,f^e=fi101 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7894795);