Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12753 |
| automorphism | Aut(G) = C2×#6628896 (order 1024) · Inn(G) = #1790 · Out(G) = C23 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #1790 |
| commutator | C2×D4 | C22×C4 |
| Frattini, cosocle | C22×D4 | C23 |
| prime radical | #3533 | C2 |
| LCS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ C22×D4 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
391 conjugacy classes of subgroups; only the 69 normal subgroups are shown.
This group has 391 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)(3,11)(4,8)(5,17,18,9)(6,10)(7,13,20,29)(12,26)(14,25)(15,23,30,21)(16,22)(19,27,31,32)(24,28);(1,3)(2,7)(4,12)(5,13)(6,14)(8,19)(9,11)(10,20)(15,27)(16,28)(17,25)(18,29)(21,26)(22,31)(23,24)(30,32);(1,4,6,16)(2,8,10,22)(3,12,14,28)(5,15,18,30)(7,19,20,31)(9,23,17,21)(11,24,25,26)(13,27,29,32)Polycyclic presentation:
2^8|a^2=f,c^2=h,b^a=bd,c^a=ce,d^a=dg,g^a=gh,e^b=egh,f^b=fg,d^c=dg,e^d=eh,f^d=fh34 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#12754, #493561, #493562, … |
| maximal normal subgroup of | C2×#12754, #493561, #493562, … |
| maximal quotient of | #129600, #129602, #129604, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5742);