Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #139978 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1889 |
| commutator | C2×C4⋊C4 | C22×C4 |
| Frattini, cosocle | C2×C4×D4 | C23 |
| prime radical | #2388 | C22 |
| LCS | G ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C2×C4×D4 ▷ C23 ▷ C2 ▷ C1 | |
564 conjugacy classes of subgroups; only the 66 normal subgroups are shown.
This group has 564 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) = 64 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2)(3,7,8,5)(4,6)(10,14)(11,17)(12,20)(16,22)(18,19)(21,23);(1,3,4,8)(2,5)(6,7)(9,10,12,14,13,16,20,22)(11,15,18,21,19,24,17,23);(2,5)(6,7)(9,11)(10,15)(12,18)(13,19)(14,21)(16,24)(17,20)(22,23)Polycyclic presentation:
2^9|a^2=f,b^2=df,d^2=gi,e^2=gi,b^a=bd,c^a=ce,d^a=dg,e^a=eghi,h^a=hi,d^b=di,e^b=ehi,f^b=fi,d^c=dh,e^c=egi,f^c=fh,e^d=ei,f^e=fi47 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 47174);