Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2190 |
| automorphism | Aut(G) = #72282918 (order 2048) · Inn(G) = #447 · Out(G) = C22×D4 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #447 |
| commutator | C2×C8 | C22×C4 |
| Frattini, cosocle | C22×C8 | C23 |
| socle | C2 | #1862 |
| prime radical | C2×#135 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C2×#97 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
194 conjugacy classes of subgroups; only the 67 normal subgroups are shown.
This group has 194 conjugacy classes of subgroups — too many to draw. Only its 67 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:
(1,2,6,12,8,13,7,26)(3,14,18,30,19,10,23,29)(4,20,17,28,21,27,16,9)(5,22,15,31,25,32,24,11);(1,3,6,18,8,19,7,23)(2,9,12,20,13,28,26,27)(4,15,17,25,21,24,16,5)(10,11,29,22,14,31,30,32);(1,4)(2,10)(3,15)(5,23)(6,17)(7,16)(8,21)(9,11)(12,29)(13,14)(18,25)(19,24)(20,22)(26,30)(27,32)(28,31)Polycyclic presentation:
2^8|a^2=f,b^2=f,d^2=gh,e^2=gh,f^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=eg,g^a=gh,d^b=dg,e^b=egh,g^b=gh,d^c=dgh,e^c=eg,g^c=gh40 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#12944, #494310, #494999, … |
| maximal normal subgroup of | C2×#12944, #494310, #494999, … |
| maximal quotient of | #135887, #135925, #136146, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5932);