Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33469 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C8 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#442 |
| commutator | C8 | C23×C8 |
| Frattini, cosocle | C4×C8 | C24 |
| prime radical | C2×#508 | C4 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ #1450 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
492 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2,4,7,5,3,6,8)(9,10,14,18,15,20,12,26)(11,21,22,38,25,37,24,17)(13,28,31,39,32,40,27,19)(16,33,35,23,36,29,34,30);(9,11)(10,16)(12,22)(13,29)(14,24)(15,25)(17,19)(18,34)(20,36)(21,40)(23,31)(26,35)(27,30)(28,37)(32,33)(38,39);(1,3,4,8,5,2,6,7)(9,11)(10,17,20,38)(12,24)(13,30)(14,22)(15,25)(16,19,36,39)(18,21,26,37)(23,32)(27,29)(28,34,40,35)(31,33);(9,12,15,14)(10,18,20,26)(11,22,25,24)(13,27,32,31)(16,34,36,35)(17,37,38,21)(19,28,39,40)(23,29,30,33)Polycyclic presentation:
2^9|a^2=f,c^2=fg,d^2=i,e^2=g,f^2=h,g^2=i,b^a=be,c^a=cg,d^a=di,e^a=eg,g^a=gi,e^b=egi,f^b=fi,g^b=gi,d^c=di,e^c=ei92 conjugacy classes — view the full character table.
| maximal subgroup of | #72216388, #72216441, #72216444, … |
| maximal normal subgroup of | #72216388, #72216441, #72216444, … |
| maximal quotient of | #72226978, #72227150, C2×#483597, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 390793);