Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #232584 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#392 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#525 | C22 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
2145 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2)(3,5)(4,6)(7,8)(9,10)(11,21)(12,26)(13,20)(14,19)(15,18)(16,23)(17,29)(22,31)(24,32)(25,33)(27,40)(28,35)(30,36)(34,37)(38,39);(5,8)(9,11,15,25)(10,16,18,32)(12,27,30,37)(13,24,14,23)(17,34,35,40)(19,33,20,21)(22,29,38,28)(26,39,36,31);(1,3)(2,5)(4,7)(6,8)(9,12,15,30)(10,17,18,35)(11,22,25,38)(13,28,14,29)(16,31,32,39)(19,36,20,26)(21,34,33,40)(23,27,24,37);(10,18)(16,32)(17,35)(19,20)(21,33)(26,36)(31,39)(34,40)Polycyclic presentation:
2^9|b^2=h,c^2=h,e^2=h,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=dh,e^a=eh,f^a=fh,g^a=ghi,c^b=cg,e^b=eh,f^b=fh,g^b=gh,e^c=ei,f^c=fh,g^c=gh62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 139783);