Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #153123 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2196 |
| commutator, Frattini, cosocle | C2×C4⋊Q8 | C23 |
| prime radical | C23 | B(4,𝔽2) |
| LCS | G ▷ C2×C4⋊Q8 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23 ◁ C22×C4 ◁ C2×C4⋊Q8 ◁ G | |
| LECS | G ▷ C2×C4⋊Q8 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
315 conjugacy classes of subgroups; only the 52 normal subgroups are shown.
This group has 315 conjugacy classes of subgroups — too many to draw. Only its 52 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:
(1,2,5,7)(3,8,4,6)(9,10)(11,15)(12,19)(13,17)(14,18)(16,20)(21,24)(22,23);(1,3,5,4)(2,6,7,8)(9,10)(11,19)(12,15)(13,23)(14,18)(16,21)(17,22)(20,24);(1,4,5,3)(2,6,7,8)(10,15)(11,20)(12,21)(16,18)(17,19)(23,24)Polycyclic presentation:
2^9|a^2=d,b^2=d,c^2=d,e^2=i,f^2=i,g^2=i,b^a=bd,c^a=ce,e^a=ei,f^a=fghi,h^a=hi,c^b=cf,e^b=eg,f^b=fi,g^b=gi,h^b=hi,e^c=ei,f^c=fi,g^c=gh,f^e=fhi,g^e=gi,g^f=gi41 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 60529);