Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13096 |
| automorphism | Aut(G) = C2×#72236702 (order 2048) · Inn(G) = #1898 · Out(G) = C24 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #1898 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C2×#41 | C23 |
| prime radical | #3010 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×2−1+4 ◁ G | |
| LECS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
320 conjugacy classes of subgroups; only the 61 normal subgroups are shown.
This group has 320 conjugacy classes of subgroups — too many to draw. Only its 61 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)(3,5,13,20)(4,14,17,26)(6,9)(7,10)(8,16,25,22)(11,28)(12,19,21,30)(15,29)(18,31)(23,27)(24,32);(1,3,5,10,7,13,20,2)(4,11,18,25,17,24,15,8)(6,21,19,9,23,12,30,27)(14,22,29,32,26,16,31,28);(1,4)(2,8)(3,11)(5,18)(6,22)(7,17)(9,26)(10,25)(12,31)(13,24)(14,27)(15,20)(16,23)(19,32)(21,29)(28,30)Polycyclic presentation:
2^8|a^2=f,b^2=dfh,e^2=h,b^a=bd,c^a=ce,d^a=dg,g^a=gh,d^b=dg,e^b=eh,f^b=fg,g^b=gh,d^c=dh,e^c=eh,f^c=fh,f^d=fh34 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#13098, #494748, #494749, … |
| maximal normal subgroup of | C2×#13098, #494748, #494749, … |
| maximal quotient of | #137686, #137689, #137733, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6086);