Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10038 |
| automorphism | Aut(G) = C2×#72278822 (order 4096) · Inn(G) = #392 · Out(G) = C2×#120 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #392 |
| commutator, Frattini, cosocle | C2×C42 | C23 |
| prime radical | #3214 | C2 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
250 conjugacy classes of subgroups; only the 49 normal subgroups are shown.
This group has 250 conjugacy classes of subgroups — too many to draw. Only its 49 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) = 128 |
A minimal faithful permutation representation — 3 generators on 32 points:
(2,7)(4,5)(8,15)(9,10)(11,16)(12,14)(17,18)(19,25)(20,28)(21,24)(22,27)(23,30)(26,32)(29,31);(1,2,6,9)(3,11,13,15)(4,10,5,7)(8,14,16,12)(22,31)(25,32);(1,3)(2,8)(4,12)(5,14)(6,13)(7,11)(9,16)(10,15)(17,19)(18,22)(20,29)(21,26)(23,32)(24,31)(25,28)(27,30)Polycyclic presentation:
2^8|b^2=g,d^2=g,e^2=h,f^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=eh,f^a=fgh,c^b=cf,d^b=dg,e^b=eg,f^b=fg,d^c=dh,e^c=eh,f^c=fg31 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#10040, #235539, #235540, … |
| maximal normal subgroup of | C2×#10040, #235539, #235540, … |
| maximal quotient of | #102471, #102475, #102481, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 3028);