Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10792 |
| automorphism | Aut(G) = #72257344 (order 2048) · Inn(G) = C4×C22⋊C4 · Out(G) = C22×D4 |
| 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 | #1749 |
| prime radical | C2×#41 | C2×C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C2×#97 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
222 conjugacy classes of subgroups; only the 97 normal subgroups are shown.
This group has 222 conjugacy classes of subgroups — too many to draw. Only its 97 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)(3,11,13,7)(4,9)(5,10)(6,8)(12,21,15,19)(14,25,28,20)(16,22)(17,24)(18,23)(26,30)(27,32,29,31);(1,3,4,14,5,15,17,29,6,13,16,28,18,12,26,27)(2,7,9,20,10,21,24,32,8,11,22,25,23,19,30,31);(3,12,13,15)(4,16)(7,19,11,21)(9,22)(14,29,28,27)(17,26)(20,32,25,31)(24,30)Polycyclic presentation:
2^8|a^2=e,b^2=f,c^2=e,f^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 | #129974, #129975, #129981, … |
| maximal normal subgroup of | #129974, #129975, #129981, … |
| maximal quotient of | #128262, #128281, #128358, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5597);