Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7304 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C8 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #342 |
| commutator | C24 | C4×C8 |
| Frattini, cosocle | C2×C8×D4 | C22 |
| socle | C22 | #1282 |
| prime radical | C22×D4 | C2×C8 |
| LCS | G ▷ C24 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C8 ◁ C2×C8×D4 ◁ G | |
| LECS | G ▷ C2×C8×D4 ▷ C22×C4 ▷ C22 ▷ C1 | |
242 conjugacy classes of subgroups; only the 58 normal subgroups are shown.
This group has 242 conjugacy classes of subgroups — too many to draw. Only its 58 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) = 128 |
A minimal faithful permutation representation — 2 generators on 24 points:
(1,2,3,6,4,5,7,8)(9,10,12,18,15,11,16,20,14,17,13,23,22,19,24,21);(2,5)(9,11,13,20,15,17,24,23,14,19,12,21,22,10,16,18)Polycyclic presentation:
2^9|a^2=d,b^2=e,d^2=g,e^2=gh,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, 200);