Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #499263 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #3109 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | C22×D8 | C23 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C4×#97 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
487 conjugacy classes of subgroups; only the 136 normal subgroups are shown.
This group has 487 conjugacy classes of subgroups — too many to draw. Only its 136 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) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,4,6)(3,7,8,5)(11,18)(14,22)(19,24)(20,23);(10,14)(11,19)(12,21)(15,20)(16,22)(17,23)(18,24);(1,3,4,8)(2,5,6,7)(9,10)(11,20)(12,15)(13,16)(14,19)(17,21)(18,23)(22,24);(10,15,16,17)(11,18)(14,20,22,23)(19,24)Polycyclic presentation:
2^9|a^2=f,c^2=f,d^2=e,g^2=hi,h^2=i,b^a=be,c^a=cf,g^a=gi,c^b=cg,g^b=gh,h^b=hi,d^c=dh,e^c=ei,g^c=gh,h^c=hi,g^d=gi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72239798, #72239808, #72239836, … |
| maximal normal subgroup of | #72239798, #72239808, #72239836, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 406825);