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 | |
4268 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,6)(3,4,8,7)(10,16)(11,12)(18,21)(19,20)(25,26,30,27)(28,33,36,31)(29,32,37,34)(35,39,40,38);(2,6)(4,7)(9,10)(11,17)(12,14)(13,21)(15,16)(18,23)(19,22)(20,24)(26,27)(31,33)(32,34)(38,39);(9,11,15,12)(10,17,16,14)(13,19,23,20)(18,24,21,22)(25,27,30,26)(28,33,36,31)(29,34,37,32)(35,39,40,38);(1,3,5,8)(2,7,6,4)(9,12,15,11)(10,17,16,14)(13,20,23,19)(18,24,21,22)(25,28)(26,31)(27,33)(29,35)(30,36)(32,38)(34,39)(37,40);(1,4)(2,3)(5,7)(6,8)(9,13,15,23)(10,18,16,21)(11,19,12,20)(14,22,17,24);(1,3)(2,4)(5,8)(6,7)(9,14,15,17)(10,12,16,11)(13,22,23,24)(18,20,21,19)(25,29,30,37)(26,32,27,34)(28,35,36,40)(31,38,33,39)Polycyclic presentation:
2^9|a^2=gh,c^2=hi,d^2=gi,e^2=h,f^2=hi,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, 8411972);