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 | |
4378 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10)(11,15)(12,19)(13,14)(16,20)(17,21)(18,24)(22,23)(25,26,31,36)(27,30,35,32)(28,33,29,34)(37,38,39,40);(1,2)(3,7)(4,5)(6,8)(11,17)(15,21)(18,23)(22,24)(25,27,31,35)(26,32,36,30)(28,37,29,39)(33,40,34,38);(1,3)(2,6)(4,7)(5,8)(10,14)(15,21)(16,19)(22,24)(25,28,31,29)(26,33,36,34)(27,37,35,39)(30,38,32,40);(9,10)(11,15)(12,16)(13,14)(17,21)(18,22)(19,20)(23,24)(25,26)(27,32)(28,33)(29,34)(30,35)(31,36)(37,40)(38,39);(1,4,5,2)(3,7,8,6)(9,11)(10,15)(12,18)(13,17)(14,21)(16,22)(19,24)(20,23)(25,29,31,28)(26,34,36,33)(27,37,35,39)(30,38,32,40);(1,3)(2,7)(4,6)(5,8)(9,12)(10,16)(11,18)(13,20)(14,19)(15,22)(17,23)(21,24)(25,30,31,32)(26,35,36,27)(28,38,29,40)(33,39,34,37)Polycyclic presentation:
2^9|a^2=g,b^2=g,c^2=g,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eghi,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, 9743126);