Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13436 |
| automorphism | Aut(G) = C2×#72236702 (order 2048) · Inn(G) = #2044 · Out(G) = C24 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #2044 |
| commutator, Frattini, cosocle | C2×#41 | C23 |
| prime radical | #535 | C22 |
| LCS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×#41 ◁ G | |
| LECS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
211 conjugacy classes of subgroups; only the 43 normal subgroups are shown.
This group has 211 conjugacy classes of subgroups — too many to draw. Only its 43 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:
(1,2,7,12)(3,13,17,30)(4,8,6,10)(5,19,20,11)(9,24,26,16)(14,28,15,29)(18,27,21,23)(22,31,25,32);(2,8)(3,14)(4,6)(5,20)(10,12)(11,27)(15,17)(16,31)(19,23)(22,25)(24,32)(28,29);(1,3,4,15,7,17,6,14)(2,9,10,25,12,26,8,22)(5,16,18,32,20,24,21,31)(11,13,27,28,19,30,23,29)Polycyclic presentation:
2^8|a^2=h,c^2=d,d^2=h,e^2=h,f^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^a=fgh,c^b=cf,d^b=dh,e^b=eg,f^b=fh,g^c=gh,f^e=fh28 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#13437, #499125, #499126, … |
| maximal normal subgroup of | C2×#13437, #499125, #499126, … |
| maximal quotient of | #144675, #144676, #144711, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6425);