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 | |
4485 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,4)(3,7,8,6)(9,10)(11,14)(12,15)(13,16)(17,19)(18,21)(20,22)(23,24)(25,26,31,28)(27,29,32,34)(30,33,39,36)(35,37,38,40);(1,3,5,8)(2,6,4,7)(10,11)(13,21)(15,17)(20,24)(25,27,31,32)(26,29,28,34)(30,38,39,35)(33,40,36,37);(2,4)(6,7)(12,19)(15,17)(20,24)(22,23)(25,26,31,28)(27,34,32,29)(30,33,39,36)(35,40,38,37);(1,3)(2,6)(4,7)(5,8)(9,11,14,10)(12,15,19,17)(13,18,21,16)(20,22,24,23)(25,28,31,26)(27,29,32,34)(30,36,39,33)(35,37,38,40);(1,4,5,2)(3,7,8,6)(9,12,14,19)(10,15,11,17)(13,20,21,24)(16,22,18,23)(25,29)(26,32)(27,28)(30,37)(31,34)(33,38)(35,36)(39,40);(1,3)(2,6)(4,7)(5,8)(9,13)(10,16)(11,18)(12,20)(14,21)(15,22)(17,23)(19,24)(25,30)(26,33)(27,35)(28,36)(29,37)(31,39)(32,38)(34,40)Polycyclic presentation:
2^9|a^2=hi,b^2=hi,c^2=i,d^2=ghi,e^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,f^b=fgi,e^c=egi,e^d=eghi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10275821);