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 | #36777 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4383 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)(4,5)(6,8)(9,10,14,17)(11,18,13,15)(12,20,22,16)(19,23,21,24)(25,26,30,27)(28,33,36,31)(29,32,37,34)(35,39,40,38);(1,3)(2,7)(4,8)(5,6)(9,11,14,13)(10,15,17,18)(12,19,22,21)(16,23,20,24)(26,27)(31,33)(32,34)(38,39);(1,4)(2,3)(5,8)(6,7)(25,27,30,26)(28,33,36,31)(29,34,37,32)(35,39,40,38);(1,5,6,3)(2,4,8,7)(9,10)(11,15)(12,16)(13,18)(14,17)(19,23)(20,22)(21,24)(25,28)(26,31)(27,33)(29,35)(30,36)(32,38)(34,39)(37,40);(9,12)(10,16)(11,19)(13,21)(14,22)(15,23)(17,20)(18,24);(1,4)(2,5)(3,8)(6,7)(9,13,14,11)(10,15,17,18)(12,21,22,19)(16,23,20,24)(25,29,30,37)(26,32,27,34)(28,35,36,40)(31,38,33,39)Polycyclic presentation:
2^9|a^2=g,b^2=ghi,c^2=hi,d^2=h,f^2=g,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, 8343775);