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×#509 | C4 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ #1450 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
509 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2,4,7,5,3,6,8)(9,10,13,19,15,20,14,31)(11,21,24,33,22,37,25,17)(12,26,27,36,30,16,29,18)(23,35,38,40,28,34,39,32);(10,16)(12,27)(13,14)(17,32)(18,31)(19,36)(20,26)(21,34)(23,38)(24,25)(28,39)(29,30)(33,40)(35,37);(1,3,4,8,5,2,6,7)(9,11)(10,17,20,33)(12,28)(13,24)(14,25)(15,22)(16,32,26,40)(18,35,36,34)(19,21,31,37)(23,30)(27,39)(29,38);(11,22)(17,33)(21,37)(23,28)(24,25)(32,40)(34,35)(38,39)Polycyclic presentation:
2^9|a^2=f,c^2=fg,e^2=g,f^2=h,g^2=i,b^a=be,c^a=cg,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 | #72216436, #72216441, #72216448, … |
| maximal normal subgroup of | #72216436, #72216441, #72216448, … |
| maximal quotient of | #72227052, #72227144, C2×#483592, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 390788);