Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12778 |
| automorphism | Aut(G) = C2×#72236702 (order 2048) · Inn(G) = #1790 · Out(G) = C24 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #1790 |
| commutator | C2×Q8 | C22×C4 |
| Frattini, cosocle | C2×#41 | C23 |
| prime radical | #3551 | C2 |
| LCS | G ▷ C2×Q8 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×#41 ◁ G | |
| LECS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
378 conjugacy classes of subgroups; only the 69 normal subgroups are shown.
This group has 378 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)(3,13)(4,17,5,10)(6,11)(7,12)(8,14,25,19)(9,18,26,20)(15,29)(16,27)(21,23)(22,28)(24,30,32,31);(1,3)(2,8)(4,14)(5,19)(6,21)(7,16)(9,23)(10,27)(11,24)(12,25)(13,17)(15,22)(18,31)(20,30)(26,29)(28,32);(1,4,7,5)(2,9,12,26)(3,14,16,19)(6,20,22,18)(8,23,25,29)(10,28,17,11)(13,24,27,32)(15,31,21,30)Polycyclic presentation:
2^8|a^2=f,c^2=h,d^2=h,e^2=h,b^a=bd,c^a=ce,d^a=dg,g^a=gh,d^b=dh,e^b=eg,f^b=fgh,d^c=dgh,e^c=eh,f^c=fh,e^d=eh,f^d=fh34 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#12780, #493620, #493621, … |
| maximal normal subgroup of | C2×#12780, #493620, #493621, … |
| maximal quotient of | #129587, #129588, #129629, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5768);