Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7304 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C8 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #342 |
| commutator | C24 | C4×C8 |
| Frattini, cosocle | C22×C22⋊C8 | C22 |
| socle | C22 | #1280 |
| prime radical | C25 | C2×C8 |
| LCS | G ▷ C24 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ C22×C22⋊C8 ◁ G | |
| LECS | G ▷ C22×C22⋊C8 ▷ C22×C4 ▷ C22 ▷ C1 | |
456 conjugacy classes of subgroups; only the 70 normal subgroups are shown.
This group has 456 conjugacy classes of subgroups — too many to draw. Only its 70 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 — 2 generators on 24 points:
(1,2)(3,7,8,5)(4,6)(9,10,12,18,14,19,20,23,15,17,21,24,13,22,11,16);(2,5)(6,7)(9,11,13,21,15,20,14,12)(10,16,19,18,17,23,22,24)Polycyclic presentation:
2^9|a^2=d,b^2=e,d^2=g,e^2=i,g^2=i,b^a=bc,c^a=cf,e^a=ei,f^a=fh,c^b=ci,d^b=df,d^c=dh68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 177);