Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12826 |
| automorphism | Aut(G) = #72311826 (order 2048) · Inn(G) = #1804 · Out(G) = C24 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #1804 |
| commutator | C2×C8 | C22×C4 |
| Frattini, cosocle | C22×C8 | C23 |
| prime radical | C2×#135 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ #405 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
195 conjugacy classes of subgroups; only the 69 normal subgroups are shown.
This group has 195 conjugacy classes of subgroups — too many to draw. Only its 69 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) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,4,10,7,12,6,24)(3,13,16,8,17,27,14,28)(5,19,18,25,23,32,22,11)(9,21,30,20,26,31,29,15);(1,3)(2,8,12,28)(4,14)(5,20)(6,16)(7,17)(9,25,26,11)(10,13,24,27)(15,23)(18,21)(19,29,32,30)(22,31);(1,3)(2,9)(4,16)(5,21)(6,14)(7,17)(8,25)(10,30)(11,28)(12,26)(13,19)(15,22)(18,20)(23,31)(24,29)(27,32)Polycyclic presentation:
2^8|a^2=f,b^2=dg,d^2=h,e^2=gh,f^2=h,g^2=h,b^a=bd,c^a=ce,e^a=eg,g^a=gh,e^b=eh,f^b=fh,d^c=dh,e^c=eg,g^c=gh40 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#12828, #493796, #494058, … |
| maximal normal subgroup of | C2×#12828, #493796, #494058, … |
| maximal quotient of | #134390, #134391, #134415, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5816);