Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7943 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#95 |
| commutator | C2×C4 | C2×C42 |
| Frattini, cosocle | C23×C4 | C23 |
| prime radical | C25 | Q8 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
437 conjugacy classes of subgroups; only the 123 normal subgroups are shown.
This group has 437 conjugacy classes of subgroups — too many to draw. Only its 123 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) = 32 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2)(3,7,8,5)(4,6)(9,10,12,17,14,18,13,16)(11,19,21,15,20,24,22,23);(1,3)(2,5)(4,8)(6,7)(9,11,13,22,14,20,12,21)(10,15,17,24,18,23,16,19);(3,8)(5,7)(11,20)(15,23)(19,24)(21,22)Polycyclic presentation:
2^8|a^2=e,b^2=f,d^2=gh,e^2=gh,f^2=gh,b^a=bd,d^a=dg,f^a=fgh,c^b=cg,e^b=eh52 conjugacy classes — view the full character table.
| maximal subgroup of | #105630, #105631, #105856, … |
| maximal normal subgroup of | #105630, #105631, #105856, … |
| maximal quotient of | #95074, #95076, #95086, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 997);