Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2202 |
| automorphism | Aut(G) = #72312045 (order 6144) · Inn(G) = B(4,𝔽2) · Out(G) = C22×S4 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | B(4,𝔽2) |
| commutator | C2×Q8 | C22×C4 |
| Frattini, cosocle | C2×#41 | C23 |
| socle | C2 | #1790 |
| prime radical | #3529 | C2 |
| LCS | G ▷ C2×Q8 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C2×C4 ◁ C2×#41 ◁ G | |
| LECS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
380 conjugacy classes of subgroups; only the 69 normal subgroups are shown.
This group has 380 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)(3,13)(4,17,5,10)(6,11)(7,12)(8,14,24,19)(9,18,25,20)(15,22)(16,26)(21,28)(23,29,27,30)(31,32);(1,3)(2,8)(4,14)(5,19)(6,15)(7,16)(9,22)(10,26)(11,27)(12,24)(13,17)(18,29)(20,30)(21,31)(23,28)(25,32);(1,4)(2,9)(3,14)(5,7)(6,20)(8,22)(10,11)(12,25)(13,23)(15,30)(16,19)(17,28)(18,21)(24,32)(26,27)(29,31)Polycyclic presentation:
2^8|a^2=f,d^2=h,e^2=h,b^a=bd,c^a=ce,d^a=dg,g^a=gh,d^b=dh,e^b=egh,f^b=fgh,d^c=dg,e^c=eh,f^c=fh,e^d=eh,f^d=fh34 conjugacy classes — view the full character table.
| Fitting subgroup of | #12772⋊C3 |
| maximal subgroup of | #152560, #152579, #152612, … |
| maximal normal subgroup of | #152560, #152579, #152612, … |
| maximal quotient of | #129581, #129582, #129623, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5760);