Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #93520 |
| automorphism | ? |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #342 |
| commutator | C23×C4 | C42 |
| Frattini, cosocle | C2×C4×C22⋊C4 | C22 |
| prime radical | C25 | C4⋊C4 |
| LCS | G ▷ C23×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ C2×C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C2×C4×C22⋊C4 ▷ C24 ▷ C2 ▷ C1 | |
436 conjugacy classes of subgroups; only the 66 normal subgroups are shown.
This group has 436 conjugacy classes of subgroups — too many to draw. Only its 66 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:
(2,5,6,7)(9,10,13,11,12,15,14,16)(18,21)(22,23);(1,2)(3,7)(4,6)(5,8)(9,11,14,15,12,16,13,10)(17,18)(19,23,24,21)(20,22)Polycyclic presentation:
2^9|a^2=d,b^2=e,c^2=fh,d^2=f,e^2=f,b^a=bc,c^a=cf,e^a=efgh,c^b=cg,d^b=dh,g^b=gi,e^c=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 768);