Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #487627 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×B(4,𝔽2) |
| commutator, Frattini, cosocle | C2×C22⋊C4 | C24 |
| prime radical | #45355 | C2 |
| LCS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
1507 conjugacy classes of subgroups; only the 120 normal subgroups are shown.
This group has 1507 conjugacy classes of subgroups — too many to draw. Only its 120 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:
(1,2,4,6)(3,5,8,7)(9,10,14,19)(11,16,21,24)(12,18,13,17)(15,22,23,20);(2,5)(3,8)(6,7)(9,11)(10,15)(12,20)(13,22)(14,21)(16,17)(18,24)(19,23);(3,8)(9,11)(10,16)(12,22)(13,20)(14,21)(15,17)(18,23)(19,24);(2,6)(5,7)(9,12,14,13)(10,17,19,18)(11,20,21,22)(15,16,23,24)Polycyclic presentation:
2^9|a^2=g,d^2=gi,e^2=g,b^a=be,c^a=cf,d^a=dg,e^a=eg,h^a=hi,e^b=eg,f^b=fhi,d^c=dgi,e^c=eh,f^e=fi50 conjugacy classes — view the full character table.
| maximal subgroup of | #72231195, #72234734, #72234738, … |
| maximal normal subgroup of | #72231195, #72234734, #72234738, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 395159);