Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #139773 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1889 |
| commutator | C22×D4 | C22×C4 |
| Frattini, cosocle | C23×D4 | C23 |
| prime radical | #3484 | C22 |
| LCS | G ▷ C22×D4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C23×D4 ◁ G | |
| LECS | G ▷ C23×D4 ▷ C23 ▷ C2 ▷ C1 | |
875 conjugacy classes of subgroups; only the 72 normal subgroups are shown.
This group has 875 conjugacy classes of subgroups — too many to draw. Only its 72 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 — 3 generators on 24 points:
(1,2)(3,5,7,8)(4,6)(9,10)(11,17,18,24)(12,15,21,23)(13,16)(14,19)(20,22);(1,3)(2,5,6,8)(4,7)(15,23)(17,24);(9,11)(10,14,16,22)(12,20)(13,18)(15,24,23,17)(19,21)Polycyclic presentation:
2^9|a^2=f,b^2=df,c^2=hi,b^a=bd,c^a=ce,d^a=dg,e^a=eh,h^a=hi,d^b=dg,e^b=ehi,f^b=fg,d^c=dh,e^c=ei,f^c=fh,e^d=ei,f^e=fi47 conjugacy classes — view the full character table.
| maximal subgroup of | #72231753, #72233078, #72233175, … |
| maximal normal subgroup of | #72231753, #72233078, #72233175, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 46970);