Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12436 |
| automorphism | Aut(G) = #72309619 (order 2048) · Inn(G) = #1667 · Out(G) = C22×C4 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #1667 |
| commutator, Frattini, cosocle | C4×C8 | C23 |
| prime radical | #2213 | C2 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C4×C8 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
112 conjugacy classes of subgroups; only the 37 normal subgroups are shown.
This group has 112 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) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(2,7)(3,12)(4,17)(5,18)(8,19)(10,24)(11,23)(13,14)(15,30)(16,25)(20,26)(21,22)(27,28)(31,32);(1,2,8,24,5,10,4,7,6,11,17,22,18,21,19,23)(3,9,25,31,14,27,13,20,16,29,12,26,15,28,30,32);(1,3)(2,9)(4,13)(5,14)(6,16)(7,20)(8,25)(10,27)(11,29)(12,17)(15,18)(19,30)(21,28)(22,26)(23,32)(24,31)Polycyclic presentation:
2^8|b^2=dfh,d^2=fh,e^2=gh,f^2=h,g^2=h,b^a=bd,c^a=ce,d^a=df,e^a=eg,f^a=fh,g^a=gh,e^c=eg,g^c=gh37 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#12438, #490484, #490485, … |
| maximal normal subgroup of | C2×#12438, #490484, #490485, … |
| maximal quotient of | #123367, #123368, #123373, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5426);