Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12364 |
| automorphism | Aut(G) = #72279473 (order 2048) · Inn(G) = #1629 · Out(G) = C22⋊C4 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #1629 |
| commutator, Frattini, cosocle | C4×C8 | C23 |
| prime radical | #3291 | 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 | |
173 conjugacy classes of subgroups; only the 39 normal subgroups are shown.
This group has 173 conjugacy classes of subgroups — too many to draw. Only its 39 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)(3,12)(4,15,17,32)(5,10)(6,20)(7,22)(8,14)(9,13,26,31)(11,19)(16,24,23,29)(18,27)(21,28,30,25);(1,3,5,14)(2,8,10,12)(4,13,17,31)(6,7,18,19)(9,24,26,29)(11,27,22,20)(15,25,32,28)(16,21,23,30);(1,4)(2,9)(3,13)(5,17)(6,21)(7,23)(8,24)(10,26)(11,28)(12,29)(14,31)(15,27)(16,19)(18,30)(20,32)(22,25)Polycyclic presentation:
2^8|a^2=f,b^2=h,d^2=f,e^2=g,g^2=h,b^a=bd,c^a=ce,d^a=df,e^a=eg,g^a=gh,d^b=df,e^b=eg,g^b=gh,d^c=dg,e^c=egh,f^c=fh,g^c=gh34 conjugacy classes — view the full character table.
| maximal subgroup of | #153638, #153639, #153641, … |
| maximal normal subgroup of | #153638, #153639, #153641, … |
| maximal quotient of | #125893, #125894, #125948, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5353);