Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #151146 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2196 |
| commutator, Frattini, cosocle | #391 | C23 |
| prime radical | #23890 | C2 |
| LCS | G ▷ #391 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23 ◁ C22×C4 ◁ #391 ◁ G | |
| LECS | G ▷ #391 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
592 conjugacy classes of subgroups; only the 40 normal subgroups are shown.
This group has 592 conjugacy classes of subgroups — too many to draw. Only its 40 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 — 3 generators on 24 points:
(1,2)(3,5)(4,7)(6,8)(9,10,14,15)(11,19,22,24)(12,23,18,13)(16,21,17,20);(2,5)(3,8)(6,7)(9,11)(10,16)(12,21)(13,24)(14,22)(15,17)(18,20)(19,23);(9,11)(10,17)(12,19)(13,20)(14,22)(15,16)(18,24)(21,23)Polycyclic presentation:
2^9|a^2=i,d^2=f,e^2=i,g^2=i,b^a=bd,c^a=ce,d^a=df,e^a=ei,g^a=gh,d^b=df,e^b=eghi,h^b=hi,d^c=dg,e^c=ei,g^c=gi,h^c=hi,e^d=ehi,g^d=gi,g^e=gi38 conjugacy classes — view the full character table.
| maximal subgroup of | #72337677, #72340681, #72340686 |
| maximal normal subgroup of | #72337677, #72340681, #72340686 |
| maximal quotient of | #72267798, #72267806, #72267807 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 58344);