Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #511796 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×B(4,𝔽2) |
| commutator, Frattini, cosocle | C22×Q8 | C24 |
| prime radical | C2×H(3,𝔽4) | C22 |
| LCS | G ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ G | |
| LECS | G ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1 | |
1039 conjugacy classes of subgroups; only the 154 normal subgroups are shown.
This group has 1039 conjugacy classes of subgroups — too many to draw. Only its 154 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)(11,14)(12,15)(13,18)(16,19)(17,21)(20,23)(22,24);(1,3,4,8)(2,5,6,7);(10,14)(11,19)(12,21)(15,23)(16,18)(17,24);(10,15)(12,21)(14,23)(16,24)(17,18)(20,22)Polycyclic presentation:
2^9|a^2=e,b^2=e,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dg,f^a=fi,g^a=gi,h^a=hi,f^c=fi,g^c=ghi,f^d=fh,g^d=gi,g^f=gi53 conjugacy classes — view the full character table.
| maximal subgroup of | Q8×#2202 |
| maximal normal subgroup of | Q8×#2202 |
| maximal quotient of | #72271531 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 419216);