Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10012 |
| automorphism | Aut(G) = C22×#72210469 (order 4096) · Inn(G) = #127 · Out(G) = C22×#120 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #127 |
| commutator, Frattini, cosocle | C2×C42 | C23 |
| socle | C22 | #392 |
| prime radical | #3100 | C2 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
275 conjugacy classes of subgroups; only the 57 normal subgroups are shown.
This group has 275 conjugacy classes of subgroups — too many to draw. Only its 57 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:
(1,2,5,3)(4,7,6,8)(9,10)(11,17)(12,20)(13,16)(14,19)(15,22)(18,24)(21,23);(2,7)(3,8)(4,6)(14,23)(17,24);(1,3)(2,5)(4,8)(6,7)(9,11)(10,14)(12,21)(13,18)(15,24)(16,23)(17,20)(19,22)Polycyclic presentation:
2^8|a^2=g,d^2=g,e^2=gh,f^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=egh,f^a=fh,c^b=cf,d^b=dg,f^b=fg,d^c=dh,e^c=egh,f^c=fg40 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#10033, #232748, #232749, … |
| maximal normal subgroup of | C2×#10033, #232748, #232749, … |
| maximal quotient of | #102430, #102438, #102442, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 3021);