Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #100923 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #386 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | C24×C4 | C23 |
| prime radical | C26 | D4 |
| LCS | G ▷ C23×C4 ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C23 ▷ C1 | |
1560 conjugacy classes of subgroups; only the 118 normal subgroups are shown.
This group has 1560 conjugacy classes of subgroups — too many to draw. Only its 118 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 — 3 generators on 24 points:
(1,2,4,7,5,8,3,6)(9,10)(11,13,15,16)(12,14)(17,18)(19,23,24,21)(20,22);(2,6,8,7)(9,11)(10,13,14,16)(12,15)(17,18)(19,21)(20,22)(23,24);(17,19)(18,21)(20,24)(22,23)Polycyclic presentation:
2^9|a^2=f,b^2=df,d^2=gh,f^2=gh,b^a=bd,c^a=ce,d^a=dg,e^a=ei,d^b=dh,e^b=ei,f^b=fh,d^c=di,f^c=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72209840, #72209853, #72209855, … |
| maximal normal subgroup of | #72209840, #72209853, #72209913, … |
| maximal quotient of | #72227746, #72227749, #72227893, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8247);