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 | |
4572 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)(3,7)(4,6)(5,8)(9,10,14,18)(11,19,21,15)(12,16,22,24)(13,17,23,20)(25,26)(27,32)(28,33)(29,39)(30,35)(31,36)(34,38)(37,40);(1,3)(2,5)(4,8)(6,7)(9,11,14,21)(10,15,18,19)(12,20,22,17)(13,16,23,24)(25,26)(27,37)(28,33)(29,34)(30,35)(31,36)(32,40)(38,39);(9,10)(11,15)(12,16)(13,17)(14,18)(19,21)(20,23)(22,24)(25,27,31,40)(26,32,36,37)(28,38,35,29)(30,39,33,34);(2,6)(5,7)(9,10)(11,19)(12,16)(13,20)(14,18)(15,21)(17,23)(22,24)(25,28)(26,33)(27,38)(29,40)(30,36)(31,35)(32,34)(37,39);(1,3)(2,5)(4,8)(6,7)(9,12,14,22)(10,16,18,24)(11,20,21,17)(13,15,23,19)(25,29,31,38)(26,34,36,39)(27,28,40,35)(30,37,33,32);(1,2)(3,5)(4,6)(7,8)(9,13)(10,17)(11,16)(12,15)(14,23)(18,20)(19,22)(21,24)(25,30,31,33)(26,35,36,28)(27,39,40,34)(29,37,38,32)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=i,e^2=hi,f^2=i,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8371566);