Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #483839 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#442 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C4×C8 | C24 |
| prime radical | C2×#509 | C4 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ #1450 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
462 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2,5,3,4,6,8,7)(9,10,13,19,15,20,14,31)(11,21,24,17,22,36,25,34)(12,26,27,38,30,16,29,18)(23,35,40,32,28,37,39,33);(2,6)(3,7)(10,16)(11,22)(12,27)(13,14)(17,33)(18,31)(19,38)(20,26)(21,35)(23,39)(28,40)(29,30)(32,34)(36,37);(1,3,5,6,4,7,8,2)(9,11)(10,17,20,34)(12,28)(13,24)(14,25)(15,22)(16,32,26,33)(18,37,38,35)(19,36,31,21)(23,30)(27,39)(29,40);(11,22)(17,34)(21,36)(23,28)(24,25)(32,33)(35,37)(39,40)Polycyclic presentation:
2^9|a^2=f,c^2=fg,e^2=g,f^2=h,g^2=i,b^a=be,c^a=cgh,e^a=eg,g^a=gi,c^b=ch,e^b=egi,f^b=fi,g^b=gi,d^c=di,e^c=ei68 conjugacy classes — view the full character table.
| maximal subgroup of | #72216730, #72216926, #72216937, … |
| maximal normal subgroup of | #72216730, #72216926, #72216937, … |
| maximal quotient of | C2×#483839, #72318471, #72318866, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 391035);