Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18224 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#120 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C2×#386 |
| prime radical | C2×#3002 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1972 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)(3,7,8,5)(4,6)(9,10)(11,20,22,31)(12,24)(13,19)(14,28)(15,18)(16,23,32,27)(17,25)(21,36,37,30)(26,34)(29,33)(35,38,40,39);(1,2)(3,5)(4,6)(7,8)(9,11,13,27,15,22,14,23)(10,16,19,31,18,32,28,20)(12,21,25,39,26,37,29,38)(17,30,34,40,33,36,24,35);(1,3)(2,5)(4,8)(6,7)(9,12)(10,17)(11,21)(13,25)(14,29)(15,26)(16,30)(18,33)(19,34)(20,35)(22,37)(23,38)(24,28)(27,39)(31,40)(32,36);(10,18)(11,22)(17,33)(19,28)(21,37)(23,27)(24,34)(38,39)Polycyclic presentation:
2^9|a^2=g,b^2=eg,e^2=h,f^2=h,b^a=be,c^a=cf,d^a=dh,f^a=fhi,d^b=dh,e^b=eh,f^b=fi,g^b=gh,e^c=ei,f^c=fh,g^c=gi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 102267);