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 | |
4418 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,8)(3,4,7,6)(9,10)(11,15)(12,13)(14,16)(17,21)(18,19)(20,22)(23,24)(25,26)(27,32)(28,31)(29,33)(30,34)(35,38)(36,39)(37,40);(1,2)(3,6)(4,7)(5,8)(9,11,14,20)(10,15,16,22)(12,18,17,23)(13,19,21,24)(25,27,31,34)(26,32,28,30)(29,37,39,35)(33,40,36,38);(1,3)(2,6)(4,8)(5,7)(10,16)(12,17)(15,22)(18,23)(26,28)(27,34)(33,36)(35,37);(1,4,5,6)(2,7,8,3)(9,12,14,17)(10,13,16,21)(11,18,20,23)(15,19,22,24)(25,28,31,26)(27,30,34,32)(29,36,39,33)(35,40,37,38);(1,3)(2,4)(5,7)(6,8)(9,11)(10,15)(12,18)(13,19)(14,20)(16,22)(17,23)(21,24)(25,29,31,39)(26,33,28,36)(27,35,34,37)(30,40,32,38);(1,2)(3,6)(4,7)(5,8)(9,13)(10,17)(11,19)(12,16)(14,21)(15,23)(18,22)(20,24)(25,30)(26,27)(28,34)(29,38)(31,32)(33,37)(35,36)(39,40)Polycyclic presentation:
2^9|a^2=g,b^2=gh,d^2=h,e^2=i,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9624956);