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 | |
4277 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)(3,5)(4,8)(6,7)(9,10,15,18)(11,16,19,23)(12,22,21,14)(13,17,20,24)(25,26,29,32)(27,30,34,38)(28,35,36,31)(33,39,40,37);(2,7)(3,8)(9,11,15,19)(10,16,18,23)(12,20,21,13)(14,17,22,24)(25,27,29,34)(26,30,32,38)(28,33,36,40)(31,37,35,39);(1,3,6,8)(2,4,7,5)(11,19)(12,21)(14,22)(16,23)(25,26)(27,30)(28,31)(29,32)(33,37)(34,38)(35,36)(39,40);(1,4,6,5)(2,3,7,8)(9,12)(10,14)(11,20)(13,19)(15,21)(16,17)(18,22)(23,24);(1,2)(3,4)(5,8)(6,7)(9,13,15,20)(10,17,18,24)(11,21,19,12)(14,23,22,16)(25,27)(26,30)(28,33)(29,34)(31,37)(32,38)(35,39)(36,40);(1,5,6,4)(2,8,7,3)(9,14)(10,12)(11,17)(13,23)(15,22)(16,20)(18,21)(19,24)(25,28,29,36)(26,31,32,35)(27,33,34,40)(30,37,38,39)Polycyclic presentation:
2^9|a^2=hi,b^2=hi,c^2=g,d^2=g,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,f^c=fgi,e^d=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8839555);