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 | C4 | C2×#442 |
| commutator | C8 | C23×C8 |
| Frattini, cosocle | C4×C8 | C24 |
| socle | C2 | C2×#1580 |
| prime radical | #3414 | C4 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C8 ◁ C4×M32 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
467 conjugacy classes of subgroups; only the 216 normal subgroups are shown.
This group has 467 conjugacy classes of subgroups — too many to draw. Only its 216 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,11,7,12,17,23,8,13,5,16,22,3,14,10)(4,15,19,29,20,9,24,27,18,28,21,26,32,25,30,31);(1,3)(2,7)(4,15)(5,10)(6,23)(8,12)(9,20)(11,14)(13,22)(16,17)(18,28)(19,26)(21,29)(24,31)(25,32)(27,30);(1,4,17,24,22,32,6,19,8,18,14,30,7,20,5,21)(2,9,10,29,3,15,16,31,13,25,23,26,12,28,11,27);(4,18)(9,25)(15,28)(19,21)(20,32)(24,30)(26,29)(27,31)Polycyclic presentation:
2^9|a^2=f,c^2=fg,e^2=g,f^2=h,g^2=i,h^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 | #72214092, #72214153, #72214568, … |
| maximal normal subgroup of | #72214092, #72214153, #72214568, … |
| maximal quotient of | #72216223, #72224944, C3×#483600, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 390796);