Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #20671 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | C2×#117 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| socle | C2 | #2283 |
| prime radical | C2×#41 | C23 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ #405 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
304 conjugacy classes of subgroups; only the 157 normal subgroups are shown.
This group has 304 conjugacy classes of subgroups — too many to draw. Only its 157 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 — 4 generators on 32 points:
(1,2)(3,12,15,8)(4,9)(5,10)(6,11)(7,14,21,29)(13,27,16,22)(17,24)(18,25)(19,26)(20,28,23,30)(31,32);(1,3)(2,7,11,21)(4,16)(5,14)(6,15)(8,10,12,26)(9,23,25,20)(13,18)(17,30)(19,29)(22,32,27,24)(28,31);(1,3)(2,8)(4,13)(5,14)(6,15)(7,10)(9,22)(11,12)(16,18)(17,28)(19,29)(20,24)(21,26)(23,32)(25,27)(30,31);(1,4,5,17,6,18,19,31)(2,9,10,24,11,25,26,32)(3,13,14,28,15,16,29,30)(7,20,12,27,21,23,8,22)Polycyclic presentation:
2^8|a^2=g,b^2=f,d^2=e,e^2=h,b^a=be,c^a=cf,f^a=fh,d^b=dh,g^b=gh,g^c=gh58 conjugacy classes — view the full character table.
| maximal subgroup of | #482709, #482710, #494016, … |
| maximal normal subgroup of | #482709, #482710, #494016, … |
| maximal quotient of | #191329, #191330, #191331, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 13776);