Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996335 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#553 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#123 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1413 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2)(3,7)(4,6)(5,8)(9,10,13,16)(11,14,18,22)(12,15,20,24)(17,21,19,23)(25,26,29,33,30,28,32,38)(27,34,36,37,35,40,39,31);(1,3)(2,5)(4,8)(6,7)(9,11)(10,14)(12,19)(13,18)(15,23)(16,22)(17,20)(21,24)(25,27)(26,31)(28,37)(29,39)(30,35)(32,36)(33,40)(34,38);(2,6)(5,7)(9,12,13,20)(10,15,16,24)(11,17,18,19)(14,21,22,23)(26,28)(27,35)(33,38)(36,39);(3,8)(5,7)(12,20)(15,24)(17,19)(21,23)(25,28,29,38,30,26,32,33)(27,34,36,37,35,40,39,31);(10,16)(11,18)(15,24)(17,19)(27,35)(31,37)(34,40)(36,39)Polycyclic presentation:
2^9|a^2=fgh,c^2=hi,d^2=fgi,f^2=i,b^a=bf,c^a=cg,e^a=ehi,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7018117);