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 | |
4413 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)(3,7)(4,8)(5,6)(9,10,15,18)(11,16,19,12)(13,14,17,24)(20,23,22,21)(25,26)(27,31)(28,32)(29,33)(30,35)(34,39)(36,38)(37,40);(1,3,6,4)(2,7,5,8)(9,11)(10,16)(12,18)(13,22)(14,21)(15,19)(17,20)(23,24)(25,27,29,35)(26,30,33,31)(28,36,37,34)(32,39,40,38);(1,3)(2,8)(4,6)(5,7)(11,19)(12,16)(13,17)(14,24)(25,27,29,35)(26,31,33,30)(28,36,37,34)(32,38,40,39);(3,4)(7,8)(9,12)(10,11)(13,23)(14,22)(15,16)(17,21)(18,19)(20,24)(25,27)(26,30)(28,36)(29,35)(31,33)(32,39)(34,37)(38,40);(1,4,6,3)(2,8,5,7)(9,13,15,17)(10,14,18,24)(11,20,19,22)(12,21,16,23)(25,28,29,37)(26,32,33,40)(27,34,35,36)(30,38,31,39);(1,5,6,2)(3,8,4,7)(9,14)(10,17)(11,21)(12,22)(13,18)(15,24)(16,20)(19,23)(25,26)(27,30)(28,32)(29,33)(31,35)(34,38)(36,39)(37,40)Polycyclic presentation:
2^9|a^2=hi,b^2=gh,c^2=g,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8970510);