Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10792 |
| automorphism | Aut(G) = #72257321 (order 1024) · Inn(G) = C4×C22⋊C4 · Out(G) = C24 |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | C4×C22⋊C4 |
| commutator | C2×C4 | C2×C42 |
| Frattini, cosocle | C22×C8 | C23 |
| socle | C2 | #1752 |
| prime radical | C2×#41 | C2×C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C2×M32 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
191 conjugacy classes of subgroups; only the 93 normal subgroups are shown.
This group has 191 conjugacy classes of subgroups — too many to draw. Only its 93 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:
(1,2,4,9,5,10,15,23,6,7,14,18,16,19,28,30)(3,8,12,21,11,22,24,32,13,17,27,29,25,20,26,31);(2,7)(3,11,13,25)(4,14)(8,20,17,22)(10,19)(12,26,27,24)(15,28)(21,32,29,31);(1,3,4,12,5,11,15,24,6,13,14,27,16,25,28,26)(2,8,9,21,10,22,23,32,7,17,18,29,19,20,30,31)Polycyclic presentation:
2^8|a^2=e,b^2=f,c^2=e,e^2=g,g^2=h,b^a=bd,d^a=dh,c^b=cg,e^b=eh,f^c=fh58 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#12618, #492984, #493226, … |
| maximal normal subgroup of | C2×#12618, #492984, #493226, … |
| maximal quotient of | #128258, #128270, #128290, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5606);