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 | |
4439 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,5)(6,8)(9,10,14,15)(11,16,18,12)(13,21,22,17)(19,20,24,23)(25,26)(27,30)(28,29)(31,32)(33,37)(34,40)(35,38)(36,39);(1,2,4,5)(3,6,7,8)(10,15)(12,16)(17,21)(20,23)(25,27,31,34)(26,30,32,40)(28,35,37,39)(29,38,33,36);(10,15)(11,18)(17,21)(19,24)(26,32)(27,34)(28,37)(36,38);(9,11)(10,16)(12,15)(13,19)(14,18)(17,23)(20,21)(22,24)(25,28)(26,33)(27,35)(29,32)(30,36)(31,37)(34,39)(38,40);(9,12,14,16)(10,11,15,18)(13,20,22,23)(17,19,21,24)(25,29,31,33)(26,28,32,37)(27,36,34,38)(30,39,40,35);(1,3,4,7)(2,6,5,8)(9,13,14,22)(10,17,15,21)(11,19,18,24)(12,20,16,23)(25,30)(26,34)(27,32)(28,36)(29,35)(31,40)(33,39)(37,38)Polycyclic presentation:
2^9|a^2=hi,b^2=gh,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^e=fi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7948362);