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 | |
4454 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)(7,8)(9,10)(11,13)(12,15)(14,20)(16,18)(17,22)(19,23)(21,24)(25,26)(27,35)(28,32)(29,33)(30,34)(31,37)(36,40)(38,39);(1,2,6,5)(3,7,4,8)(9,11)(10,16)(12,13)(14,19)(15,18)(17,23)(20,21)(22,24)(25,27,30,37)(26,31,34,35)(28,29,36,39)(32,38,40,33);(1,2,6,5)(3,8,4,7)(10,12)(11,18)(17,20)(19,24);(1,3,6,4)(2,7,5,8)(9,12,15,10)(11,13,18,16)(14,20,22,17)(19,21,24,23)(25,28,30,36)(26,32,34,40)(27,29,37,39)(31,38,35,33);(1,4,6,3)(2,7,5,8)(9,13)(10,11)(12,18)(14,21)(15,16)(17,19)(20,24)(22,23)(25,29)(26,33)(27,36)(28,37)(30,39)(31,32)(34,38)(35,40);(1,5,6,2)(3,8,4,7)(9,14,15,22)(10,17,12,20)(11,19,18,24)(13,21,16,23)(25,26)(27,31)(28,32)(29,38)(30,34)(33,39)(35,37)(36,40)Polycyclic presentation:
2^9|b^2=gi,c^2=hi,d^2=g,e^2=hi,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9637878);