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 | C22×D16 | C4 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ #1450 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
526 conjugacy classes of subgroups; only the 218 normal subgroups are shown.
This group has 526 conjugacy classes of subgroups — too many to draw. Only its 218 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| 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)(9,10,13,19,14,17,30,20)(11,21,24,16,22,34,37,35)(12,25,28,18,29,15,26,32)(23,36,39,31,27,33,38,40);(10,15)(12,26)(13,30)(16,31)(17,25)(18,19)(20,32)(21,33)(23,38)(24,37)(27,39)(28,29)(34,36)(35,40);(1,3,4,8,5,2,6,7)(9,11)(10,16,17,35)(12,27)(13,24)(14,22)(15,31,25,40)(18,36,32,33)(19,34,20,21)(23,29)(26,39)(28,38)(30,37);(10,17)(11,22)(15,25)(18,32)(19,20)(23,27)(24,37)(38,39)Polycyclic presentation:
2^9|a^2=f,c^2=fgi,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 | #72216421, #72216441, #72216456, … |
| maximal normal subgroup of | #72216421, #72216441, #72216456, … |
| maximal quotient of | #72227100, #72227128, C2×#483596, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 390792);