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 | |
4483 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,6)(3,4,8,7)(12,20)(16,21)(19,24)(22,23)(26,31)(27,33)(34,37)(35,40);(2,6)(3,8)(9,10)(11,15)(12,21)(13,17)(14,18)(16,20)(19,22)(23,24)(25,26,30,31)(27,32,33,28)(29,34,38,37)(35,39,40,36);(9,11)(10,13)(12,19)(14,17)(15,18)(16,22)(20,24)(21,23)(25,27,30,33)(26,32,31,28)(29,35,38,40)(34,39,37,36);(2,6)(4,7)(9,11)(10,15)(12,19)(13,18)(14,17)(16,23)(20,24)(21,22)(25,26)(27,32)(28,33)(29,37)(30,31)(34,38)(35,36)(39,40);(1,3,5,8)(2,7,6,4)(9,12)(10,16)(11,19)(13,22)(14,20)(15,23)(17,24)(18,21)(25,28,30,32)(26,33,31,27)(29,36,38,39)(34,40,37,35);(1,4,5,7)(2,8,6,3)(9,13,14,15)(10,17,18,11)(12,22,20,23)(16,24,21,19)(25,29,30,38)(26,34,31,37)(27,35,33,40)(28,36,32,39)Polycyclic presentation:
2^9|a^2=gh,b^2=h,c^2=h,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,e^b=egi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8403780);