Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18224 |
| automorphism | Aut(G) = C22×#1202 (order 4096) · Inn(G) = C2×#120 · Out(G) = C23×D4 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | #3002 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
570 conjugacy classes of subgroups; only the 113 normal subgroups are shown.
This group has 570 conjugacy classes of subgroups — too many to draw. Only its 113 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:
(5,8)(9,10)(11,18)(12,15)(13,17)(14,22)(16,20)(19,23)(21,24);(1,2)(3,5)(4,6)(7,8)(9,11)(10,16)(12,19)(13,20)(14,23)(15,21)(17,24)(18,22);(1,3)(2,5)(4,7)(6,8)(9,12,15,10)(11,19,21,16)(13,17,14,22)(18,20,24,23);(9,13,15,14)(10,17,12,22)(11,20,21,23)(16,24,19,18)Polycyclic presentation:
2^8|c^2=g,d^2=g,e^2=g,b^a=be,c^a=cf,e^a=eg,e^b=eg,f^b=fh,d^c=dg,e^c=eh37 conjugacy classes — view the full character table.
| maximal subgroup of | #249883, #249885, #249888, … |
| maximal normal subgroup of | #249883, #249885, #249888, … |
| maximal quotient of | #166294, #166311, #166350, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 11218);