Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #145145 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2034 |
| commutator, Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | #28231 | C2 |
| LCS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C2×#436 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
403 conjugacy classes of subgroups; only the 78 normal subgroups are shown.
This group has 403 conjugacy classes of subgroups — too many to draw. Only its 78 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:
(2,5)(6,7)(10,14)(11,17)(12,20)(15,22)(16,19)(18,24)(21,23);(1,2)(3,5)(4,6)(7,8)(11,18)(14,22)(17,24)(19,23);(2,6)(4,8)(5,7)(9,10,12,21,13,15,20,16)(11,19,17,22,18,23,24,14)Polycyclic presentation:
2^9|c^2=dgh,e^2=gi,f^2=h,g^2=i,b^a=bd,c^a=ce,e^a=eg,g^a=gi,c^b=cf,e^b=ei,f^b=fh,d^c=di,e^c=egi,f^c=fh,g^c=gi50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 52358);