Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1665 |
| automorphism | Aut(G) = C25×D42 (order 2048) · Inn(G) = #121 · Out(G) = C26 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #121 |
| commutator | C42 | C22×C4 |
| Frattini, cosocle | C2×C42 | C23 |
| socle | C22 | #388 |
| prime radical | #532 | C22 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
182 conjugacy classes of subgroups; only the 65 normal subgroups are shown.
This group has 182 conjugacy classes of subgroups — too many to draw. Only its 65 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:
(2,3)(4,5)(7,8)(9,10,12,14,13,17,21,16)(11,18,19,24,20,15,23,22);(1,2,5,7,6,8,4,3)(10,14,17,16)(15,22,18,24);(1,3,4,8,6,7,5,2)(9,11,12,19,13,20,21,23)(10,15,16,24,17,18,14,22)Polycyclic presentation:
2^8|a^2=f,b^2=dfg,c^2=d,d^2=g,e^2=h,f^2=gh,b^a=bd,c^a=ce,d^a=dg,e^a=eh,e^c=egh58 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#9590, #203118, #203122, … |
| maximal normal subgroup of | C2×#9590, #203118, #203122, … |
| maximal quotient of | #101897, #101898, #101899, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 2578);