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 | #36713 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4438 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,14,18)(11,13,17,15)(12,16,21,23)(19,22,24,20)(25,26)(27,35)(28,39)(29,40)(30,34)(31,32)(33,36)(37,38);(1,2)(3,7)(4,6)(5,8)(9,11,14,17)(10,15,18,13)(12,19,21,24)(16,20,23,22)(26,32)(27,36)(28,40)(34,38);(1,3)(2,5)(4,8)(6,7)(25,27,31,36)(26,33,32,35)(28,37,40,30)(29,38,39,34);(2,6)(5,7)(9,11)(10,15)(12,19)(13,18)(14,17)(16,20)(21,24)(22,23)(25,28)(26,29)(27,37)(30,36)(31,40)(32,39)(33,38)(34,35);(1,3)(2,7)(4,8)(5,6)(9,12)(10,16)(11,19)(13,22)(14,21)(15,20)(17,24)(18,23)(25,29)(26,28)(27,38)(30,35)(31,39)(32,40)(33,37)(34,36);(1,2)(3,5)(4,6)(7,8)(9,13,14,15)(10,17,18,11)(12,20,21,22)(16,19,23,24)(25,30,31,37)(26,34,32,38)(27,28,36,40)(29,35,39,33)Polycyclic presentation:
2^9|a^2=gh,b^2=gh,c^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^d=fgi,f^e=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8596439);