Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1619 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #120 |
| commutator | C42 | C22×C4 |
| Frattini, cosocle | C2×C42 | C23 |
| socle | C22 | #385 |
| prime radical | #532 | C22 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
206 conjugacy classes of subgroups; only the 63 normal subgroups are shown.
This group has 206 conjugacy classes of subgroups — too many to draw. Only its 63 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 — 3 generators on 24 points:
(2,5)(3,7)(6,8)(9,10,13,17,14,12,21,16)(11,15,19,23,20,18,24,22);(1,2,4,6)(3,8,7,5)(9,11,13,19,14,20,21,24)(10,15,17,23,12,18,16,22);(3,7)(5,8)(9,12)(10,14)(11,18)(13,16)(15,20)(17,21)(19,22)(23,24)Polycyclic presentation:
2^8|a^2=f,b^2=fg,d^2=g,e^2=h,f^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=eh,d^b=dg,e^b=eg,d^c=dg,e^c=eh52 conjugacy classes — view the full character table.
| maximal subgroup of | #94598, #152467, #153280, … |
| maximal normal subgroup of | #94598, #152467, #153280, … |
| maximal quotient of | #99338, #99345, #99346, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 1923);