Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8465 |
| automorphism | Aut(G) = C23×D43 (order 4096) · Inn(G) = #379 · Out(G) = C26 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #379 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C2×C42 | C23 |
| prime radical | C2×#41 | C2×C4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
366 conjugacy classes of subgroups; only the 79 normal subgroups are shown.
This group has 366 conjugacy classes of subgroups — too many to draw. Only its 79 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) = 128 |
A minimal faithful permutation representation — 3 generators on 24 points:
(2,5,6,7)(9,10,13,18,14,17,21,24)(11,16,19,23,12,15,20,22);(1,2)(3,7)(4,6)(5,8)(9,11,14,12)(10,15,17,16)(13,20,21,19)(18,23,24,22);(3,8)(5,7)(9,12,14,11)(10,16,17,15)(13,19,21,20)(18,22,24,23)Polycyclic presentation:
2^8|a^2=f,b^2=g,c^2=g,d^2=h,f^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=eg,d^b=dh,e^b=eh,f^b=fgh,d^c=dh,f^c=fg40 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#8473, #187034, #187037, … |
| maximal normal subgroup of | C2×#8473, #187034, #187037, … |
| maximal quotient of | #97329, #97332, #97336, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 1461);