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 |
| prime radical | #2407 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4323 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,6)(3,4,7,8)(9,10,14,18)(11,15,16,12)(13,17,22,21)(19,23,24,20)(25,26,30,29)(27,33,34,31)(28,32,36,37)(35,39,40,38);(2,6)(3,7)(9,10)(11,15)(12,16)(13,17)(14,18)(19,23)(20,24)(21,22)(27,34)(28,36)(31,33)(32,37);(1,3)(2,4)(5,7)(6,8)(9,11,14,16)(10,15,18,12)(13,19,22,24)(17,23,21,20)(25,27,30,34)(26,31,29,33)(28,35,36,40)(32,38,37,39);(1,4)(2,3)(5,8)(6,7)(9,10)(11,15)(12,16)(13,21)(14,18)(17,22)(19,20)(23,24)(25,28)(26,32)(27,35)(29,37)(30,36)(31,38)(33,39)(34,40);(9,12,14,15)(10,16,18,11)(13,20,22,23)(17,24,21,19);(9,13,14,22)(10,17,18,21)(11,19,16,24)(12,20,15,23)(25,29,30,26)(27,33,34,31)(28,32,36,37)(35,38,40,39)Polycyclic presentation:
2^9|a^2=gh,c^2=gi,e^2=ghi,f^2=gi,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, 8371530);