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 | #2537 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4346 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,5)(6,8)(9,10,13,14)(11,17,18,15)(12,16,22,20)(19,24,21,23)(26,32)(28,38)(33,35)(36,39);(1,2)(3,6)(4,5)(7,8)(9,10)(11,15)(12,20)(13,14)(16,22)(17,18)(19,24)(21,23)(25,26,31,32)(27,35,30,33)(28,34,38,29)(36,40,39,37);(10,14)(11,18)(16,20)(19,21)(25,27,31,30)(26,33,32,35)(28,36,38,39)(29,40,34,37);(9,10)(11,15)(12,16)(13,14)(17,18)(19,23)(20,22)(21,24)(25,28)(26,34)(27,36)(29,32)(30,39)(31,38)(33,37)(35,40);(1,3,4,7)(2,6,5,8)(9,11,13,18)(10,15,14,17)(12,21,22,19)(16,24,20,23)(25,29,31,34)(26,28,32,38)(27,37,30,40)(33,39,35,36);(9,12,13,22)(10,16,14,20)(11,19,18,21)(15,23,17,24)(25,30,31,27)(26,35,32,33)(28,39,38,36)(29,37,34,40)Polycyclic presentation:
2^9|a^2=h,b^2=hi,c^2=hi,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, 7893279);