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×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #342 |
| commutator | C24 | C4×C8 |
| Frattini, cosocle | C22×C4×D4 | C22 |
| prime radical | C23×D4 | C2×C4 |
| LCS | G ▷ C24 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ C22×C4×D4 ◁ G | |
| LECS | G ▷ C22×C4×D4 ▷ C24 ▷ C2 ▷ C1 | |
688 conjugacy classes of subgroups; only the 82 normal subgroups are shown.
This group has 688 conjugacy classes of subgroups — too many to draw. Only its 82 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)(3,7,8,5)(4,6)(9,10,11,13,12,14,15,16)(17,18,21,24,22,19,20,23);(1,3)(2,5)(4,8)(6,7)(17,19,22,18)(20,24)(21,23)Polycyclic presentation:
2^9|a^2=d,b^2=e,d^2=h,b^a=bc,c^a=cf,e^a=eg,f^a=fi,g^a=gi,c^b=cg,d^b=df,f^b=fi,g^b=gi,d^c=di,e^c=ei,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 678);