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 | #55935 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4178 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)(3,4,8,7)(9,10,15,14)(11,16,21,20)(12,22,17,13)(18,24,23,19)(25,26,30,27)(28,33,36,31)(29,32,37,34)(35,39,40,38);(2,6)(4,7)(9,11,15,21)(10,16,14,20)(12,18,17,23)(13,19,22,24)(26,27)(31,33)(32,34)(38,39);(9,12)(10,13)(11,18)(14,22)(15,17)(16,19)(20,24)(21,23)(25,27,30,26)(28,33,36,31)(29,34,37,32)(35,39,40,38);(1,3,5,8)(2,7,6,4)(9,11)(10,16)(12,18)(13,19)(14,20)(15,21)(17,23)(22,24)(25,28)(26,31)(27,33)(29,35)(30,36)(32,38)(34,39)(37,40);(1,4)(2,3)(5,7)(6,8)(9,13,15,22)(10,17,14,12)(11,19,21,24)(16,23,20,18);(1,3)(2,4)(5,8)(6,7)(9,14,15,10)(11,20,21,16)(12,22,17,13)(18,24,23,19)(25,29,30,37)(26,32,27,34)(28,35,36,40)(31,38,33,39)Polycyclic presentation:
2^9|a^2=gh,b^2=h,c^2=i,d^2=gi,e^2=h,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,e^b=egi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8411934);