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 | |
4740 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)(3,7)(4,6)(5,8)(9,10,15,16)(11,13,12,19)(14,22,23,17)(18,24,20,21)(25,26)(27,30)(28,31)(29,32)(33,34)(35,37)(36,38)(39,40);(1,2,4,6)(3,5,8,7)(9,11,15,12)(10,13,16,19)(14,18,23,20)(17,21,22,24)(25,26,29,32)(27,30,33,34)(28,31,36,38)(35,37,39,40);(1,2)(3,5)(4,6)(7,8)(10,16)(11,12)(14,23)(21,24)(27,33)(28,36)(30,34)(31,38);(1,3)(2,5)(4,8)(6,7)(9,12,15,11)(10,13,16,19)(14,20,23,18)(17,21,22,24)(25,27,29,33)(26,30,32,34)(28,35,36,39)(31,37,38,40);(9,13)(10,12)(11,16)(14,21)(15,19)(17,20)(18,22)(23,24)(25,26)(27,34)(28,31)(29,32)(30,33)(35,40)(36,38)(37,39);(9,14,15,23)(10,17,16,22)(11,18,12,20)(13,21,19,24)(25,28)(26,31)(27,35)(29,36)(30,37)(32,38)(33,39)(34,40)Polycyclic presentation:
2^9|a^2=h,b^2=gh,d^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, 8837591);