Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12324 |
| automorphism | Aut(G) = C23×#21452 (order 2048) · Inn(G) = #447 · Out(G) = C23×C4 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #447 |
| commutator, Frattini, cosocle | C4×C8 | C23 |
| prime radical | #3403 | C2 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ #446 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
151 conjugacy classes of subgroups; only the 39 normal subgroups are shown.
This group has 151 conjugacy classes of subgroups — too many to draw. Only its 39 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,5,8)(3,9,10,16)(4,12,13,7)(6,14,15,11)(18,23)(19,22)(20,27)(24,31)(25,32)(26,29)(28,30);(4,13)(7,12)(9,16)(11,14)(17,18)(19,24)(20,28)(21,25)(22,30)(23,27)(26,32)(29,31);(1,3,5,10)(2,6,8,15)(4,14,13,11)(7,16,12,9)(17,19,21,22)(18,24,25,30)(20,27,29,26)(23,31,32,28)Polycyclic presentation:
2^8|a^2=gh,c^2=g,d^2=fh,e^2=gh,f^2=h,b^a=bd,c^a=ce,d^a=df,e^a=egh,f^a=fh,d^b=df,e^b=eg,f^b=fh,d^c=dg,e^c=egh31 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#12390, #489237, #489238, … |
| maximal normal subgroup of | C2×#12390, #489237, #489238, … |
| maximal quotient of | #125961, #125962, #125995, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5378);