Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7213 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C8 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #342 |
| commutator | C22×C4 | C4×C8 |
| Frattini, cosocle | C2×C4×C22⋊C4 | C22 |
| prime radical | C25 | C4⋊C4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ C2×C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C2×C4×C22⋊C4 ▷ C24 ▷ C2 ▷ C1 | |
454 conjugacy classes of subgroups; only the 76 normal subgroups are shown.
This group has 454 conjugacy classes of subgroups — too many to draw. Only its 76 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)(3,5)(4,7)(6,8)(9,10,12,15,14,16,13,11)(17,18)(19,23,24,21)(20,22);(2,5,7,6)(9,11,13,16,14,15,12,10)(18,21)(22,23)Polycyclic presentation:
2^9|a^2=d,b^2=e,c^2=fg,d^2=h,e^2=h,b^a=bc,c^a=cf,e^a=efg,f^a=fi,g^a=gi,d^b=dg,d^c=di80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 537);