Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #353 |
| automorphism | Aut(G) = C22×D4×B(4,𝔽2) (order 2048) · Inn(G) = #99 · Out(G) = C23×D4 |
| homology | Abelianization (H1) = C4×C8 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #99 |
| commutator | C2×C4 | C4×C8 |
| Frattini, cosocle | C2×C4×Q8 | C22 |
| socle | C22 | #342 |
| prime radical | C23 | #95 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C4 ◁ C2×C4×Q8 ◁ G | |
| LECS | G ▷ C2×C4×Q8 ▷ C23 ▷ C2 ▷ C1 | |
142 conjugacy classes of subgroups; only the 51 normal subgroups are shown.
This group has 142 conjugacy classes of subgroups — too many to draw. Only its 51 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 — 2 generators on 24 points:
(1,2,4,7,6,8,5,3)(9,10,12,16,13,17,21,24)(11,15,18,23,19,14,20,22);(1,3,5,8,6,7,4,2)(10,14,17,15)(16,23,24,22)Polycyclic presentation:
2^8|a^2=d,b^2=e,c^2=h,d^2=g,e^2=gh,b^a=bc,c^a=cf,e^a=eh,f^a=fh,d^b=dfh,d^c=dh52 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#7206, #127078, #127079, … |
| maximal normal subgroup of | C2×#7206, #127078, #127079, … |
| maximal quotient of | #92998, #93002, #93006, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 194);