Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2184 |
| automorphism | Aut(G) = C2×#72237167 (order 4096) · Inn(G) = C2×D8 · Out(G) = C2×#525 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×D8 |
| commutator | C2×C8 | C22×C4 |
| Frattini, cosocle | C4×C8 | C23 |
| socle | C22 | C2×#102 |
| prime radical | #466 | C4 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
160 conjugacy classes of subgroups; only the 61 normal subgroups are shown.
This group has 160 conjugacy classes of subgroups — too many to draw. Only its 61 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:
(1,2,4,7,5,8,11,14)(3,6,10,13,9,12,15,16)(18,22)(19,25)(20,27)(23,30)(24,31)(26,29)(28,32);(1,3)(2,6)(4,10)(5,9)(7,13)(8,12)(11,15)(14,16)(17,18)(19,26)(20,29)(21,23)(22,25)(24,27)(28,31)(30,32);(3,9)(6,12)(10,15)(13,16)(18,23)(22,30)(24,29)(26,31)Polycyclic presentation:
2^8|a^2=e,d^2=fh,e^2=g,f^2=h,b^a=bd,d^a=df,f^a=fh,c^b=cgh,d^b=df,f^b=fh58 conjugacy classes — view the full character table.
| maximal subgroup of | #131624, #131625, #131634, … |
| maximal normal subgroup of | #131624, #131625, #131634, … |
| maximal quotient of | #125080, #125082, #125128, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4636);