Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #511785 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×B(4,𝔽2) |
| commutator, Frattini, cosocle | C22×D4 | C24 |
| prime radical | C2×H(3,𝔽4) | C22 |
| LCS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
1047 conjugacy classes of subgroups; only the 152 normal subgroups are shown.
This group has 1047 conjugacy classes of subgroups — too many to draw. Only its 152 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 — 4 generators on 24 points:
(1,2,4,6)(3,7,8,5)(9,10,12,16)(11,18,19,14)(13,17,21,22)(15,24,23,20);(1,3,4,8)(2,5,6,7)(9,11,12,19)(10,14,16,18)(13,20,21,24)(15,17,23,22);(9,11)(10,15)(12,19)(13,20)(14,22)(16,23)(17,18)(21,24);(11,19)(13,21)(14,18)(15,23)Polycyclic presentation:
2^9|a^2=e,b^2=e,b^a=be,c^a=cf,d^a=dg,h^a=hi,c^b=ci,d^b=di,d^c=di,g^c=ghi,f^d=fh,g^f=gi53 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 419213);