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 | |
4191 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,4,6)(3,5,7,8)(9,10,15,17)(11,19,13,16)(12,14,20,18)(21,22,23,24)(25,26,31,36)(27,29,34,32)(28,33,30,35)(37,40,38,39);(1,2)(3,5)(4,6)(7,8)(9,11,15,13)(10,16,17,19)(12,21,20,23)(14,24,18,22)(25,27,31,34)(26,32,36,29)(28,38,30,37)(33,40,35,39);(10,17)(11,13)(12,20)(22,24)(25,28,31,30)(26,33,36,35)(27,37,34,38)(29,40,32,39);(9,12,15,20)(10,18,17,14)(11,21,13,23)(16,22,19,24)(25,26)(27,32)(28,33)(29,34)(30,35)(31,36)(37,39)(38,40);(1,3,4,7)(2,5,6,8)(9,13,15,11)(10,19,17,16)(12,23,20,21)(14,22,18,24)(25,29,31,32)(26,34,36,27)(28,39,30,40)(33,37,35,38);(9,14,15,18)(10,20,17,12)(11,22,13,24)(16,21,19,23)(25,30,31,28)(26,35,36,33)(27,38,34,37)(29,39,32,40)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=hi,d^2=h,e^2=g,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, 7893298);