Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #13603 |
| automorphism | Aut(G) = #72317530 (order 2048) · Inn(G) = C2×#354 · Out(G) = C24 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C2×#354 |
| commutator | C42 | C22×C4 |
| Frattini, cosocle | #124 | C23 |
| prime radical | C22×D4 | D4 |
| LCS | G ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ C22×C4 ◁ C2×#124 ◁ G | |
| LECS | G ▷ #124 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
167 conjugacy classes of subgroups; only the 45 normal subgroups are shown.
This group has 167 conjugacy classes of subgroups — too many to draw. Only its 45 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)(3,11,14,27)(4,15,17,31)(5,19,21,9)(6,10)(7,13,23,29)(8,18)(12,26,20,28)(16,24,30,22)(25,32);(1,3)(2,7,10,22)(4,16)(5,20)(6,12)(8,11,25,26)(9,23,19,24)(13,17)(14,21)(15,27,31,28)(18,30)(29,32);(3,12)(7,22)(11,26)(13,16)(14,20)(23,24)(27,28)(29,30)Polycyclic presentation:
2^8|a^2=e,b^2=g,d^2=g,f^2=h,b^a=bd,d^a=df,f^a=fg,g^a=gh,c^b=ch,d^b=dgh,e^b=efgh,f^b=fh,e^d=egh,f^e=fh28 conjugacy classes — view the full character table.
| maximal subgroup of | #152071, #152072, #152078, … |
| maximal normal subgroup of | #152071, #152072, #152078, … |
| maximal quotient of | #149587, #149611, #149706, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6591);