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 | |
4683 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)(3,4)(5,8)(6,7)(9,10)(11,16)(12,15)(13,17)(14,19)(18,22)(20,23)(21,24)(25,26)(27,35)(28,32)(29,39)(30,34)(31,36)(33,37)(38,40);(1,3,5,7)(2,4,8,6)(9,11)(10,14)(12,18)(13,19)(15,21)(16,17)(20,24)(22,23)(25,27,30,36)(26,31,34,35)(28,37,38,29)(32,39,40,33);(1,3)(2,6)(4,8)(5,7)(25,26,30,34)(27,31,36,35)(28,32,38,40)(29,33,37,39);(1,2,5,8)(3,4,7,6)(9,12,13,20)(10,15,17,23)(11,18,19,24)(14,21,16,22)(25,26)(27,31)(28,32)(29,33)(30,34)(35,36)(37,39)(38,40);(1,3)(2,4)(5,7)(6,8)(9,11)(10,16)(12,18)(13,19)(14,17)(15,22)(20,24)(21,23)(25,28,30,38)(26,32,34,40)(27,29,36,37)(31,33,35,39);(1,4)(2,7)(3,8)(5,6)(9,10)(11,14)(12,15)(13,17)(16,19)(18,21)(20,23)(22,24)(25,29)(26,33)(27,28)(30,37)(31,32)(34,39)(35,40)(36,38)Polycyclic presentation:
2^9|b^2=h,c^2=hi,d^2=gh,e^2=hi,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, 9633588);