Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10018 |
| automorphism | Aut(G) = C2×#72278229 (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 | #3109 | C2 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
270 conjugacy classes of subgroups; only the 49 normal subgroups are shown.
This group has 270 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) = 64 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2)(3,11)(4,7)(5,9)(6,10)(8,12)(13,15)(14,16)(18,22)(20,27)(23,31)(24,28)(25,29)(30,32);(2,7)(4,5)(8,11)(9,10)(12,14)(15,16)(17,18)(19,23)(20,28)(21,24)(22,27)(25,30)(26,32)(29,31);(1,3,6,13)(2,8,10,16)(4,12,5,14)(7,15,9,11)(17,19,21,26)(18,23,24,32)(20,29,27,25)(22,30,28,31)Polycyclic presentation:
2^8|c^2=g,d^2=g,e^2=gh,b^a=bd,c^a=ce,d^a=dg,e^a=egh,f^a=fgh,c^b=cf,d^b=dg,e^b=eg,d^c=dh,e^c=egh31 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#10019, #235431, #235432, … |
| maximal normal subgroup of | C2×#10019, #235431, #235432, … |
| maximal quotient of | #103007, #103009, #103010, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 3007);