Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #213828 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2308 |
| commutator | C24 | C23×C4 |
| Frattini, cosocle | C25 | C24 |
| prime radical | #35346 | C2 |
| LCS | G ▷ C24 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C25 ◁ G | |
| LECS | G ▷ C25 ▷ C22 ▷ C1 | |
2310 conjugacy classes of subgroups; only the 206 normal subgroups are shown.
This group has 2310 conjugacy classes of subgroups — too many to draw. Only its 206 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,7,8,5)(4,6)(9,10)(11,17,19,14)(12,20,21,15)(13,16)(18,22)(23,24);(9,11,13,19)(10,14,16,17)(12,18,21,24)(15,22,20,23);(1,3)(2,5)(4,8)(6,7)(14,17)(22,23);(9,12)(10,15)(11,18)(13,21)(14,22)(16,20)(17,23)(19,24)Polycyclic presentation:
2^9|a^2=g,b^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,c^b=ce,f^b=fh,g^b=gh,g^c=gi,g^d=gh56 conjugacy classes — view the full character table.
| maximal subgroup of | #72225776, #72226351, #72234627, … |
| maximal normal subgroup of | #72225776, #72226351, #72234627, … |
| maximal quotient of | #72241167, #72251430, #72251455, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 121025);