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 | |
4366 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)(11,14)(12,19)(13,16)(15,20)(17,21)(18,24)(22,23)(25,26,31,29)(27,35,36,32)(28,37,38,33)(30,34,40,39);(1,2)(3,7)(4,5)(6,8)(11,17)(14,21)(18,23)(22,24)(25,27,31,36)(26,32,29,35)(28,34,38,39)(30,33,40,37);(1,3)(2,6)(4,7)(5,8)(9,10,13,16)(11,14,17,21)(12,15,20,19)(18,22,23,24)(25,26,31,29)(27,32,36,35)(28,33,38,37)(30,34,40,39);(9,10)(11,14)(12,15)(13,16)(17,21)(18,22)(19,20)(23,24)(25,28,31,38)(26,33,29,37)(27,34,36,39)(30,32,40,35);(1,4,5,2)(3,7,8,6)(9,11)(10,14)(12,18)(13,17)(15,22)(16,21)(19,24)(20,23)(25,29,31,26)(27,32,36,35)(28,37,38,33)(30,34,40,39);(1,3)(2,7)(4,6)(5,8)(9,12)(10,15)(11,18)(13,20)(14,22)(16,19)(17,23)(21,24)(25,30,31,40)(26,34,29,39)(27,33,36,37)(28,32,38,35)Polycyclic presentation:
2^9|a^2=g,b^2=g,c^2=gh,d^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, 9743326);