Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7213 |
| automorphism | Aut(G) = C22×#6499892 (order 2048) · Inn(G) = #342 · Out(G) = C25 |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #342 |
| commutator | C22×C4 | C42 |
| Frattini, cosocle | C2×#117 | C22 |
| prime radical | C24 | C22⋊C4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C2×#117 ▷ C23 ▷ C2 ▷ C1 | |
145 conjugacy classes of subgroups; only the 37 normal subgroups are shown.
This group has 145 conjugacy classes of subgroups — too many to draw. Only its 37 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 2 generators on 40 points:
(1,2)(3,5)(4,7)(6,8)(9,10)(11,19)(12,23,24,16)(13,17)(14,18)(15,20,27,33)(21,31)(22,32)(25,36,37,29)(26,30)(28,34,38,40)(35,39);(2,5,7,6)(9,11,13,21,14,22,26,35)(10,15,17,28,18,27,30,38)(12,20,25,34,24,33,37,40)(16,19,29,31,23,32,36,39)Polycyclic presentation:
2^8|a^2=d,b^2=e,c^2=fg,e^2=h,b^a=bc,c^a=cf,e^a=efg,f^a=fh,g^a=gh,d^b=dg,d^c=dh40 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#7214, #127188, #127189, … |
| maximal normal subgroup of | C2×#7214, #127188, #127189, … |
| maximal quotient of | #93301, #93342, #93368, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 202);