Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2048 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #394 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C23×C4 | C23 |
| socle | C2 | #1439 |
| prime radical | C22×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
432 conjugacy classes of subgroups; only the 117 normal subgroups are shown.
This group has 432 conjugacy classes of subgroups — too many to draw. Only its 117 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)(3,13)(4,17,20,9)(5,12)(6,10)(7,11)(8,15)(14,29,31,25)(16,27)(18,32,30,23)(19,26,22,28)(21,24);(1,3,5,15,7,16,6,21)(2,8,10,13,11,24,12,27)(4,18,19,14,20,30,22,31)(9,25,26,23,17,29,28,32);(1,4)(2,9)(3,14)(5,19)(6,22)(7,20)(8,23)(10,26)(11,17)(12,28)(13,29)(15,30)(16,31)(18,21)(24,32)(25,27)Polycyclic presentation:
2^8|a^2=g,b^2=d,d^2=h,f^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^a=fh,c^b=cf,e^b=eh,f^b=fh,f^c=fh,g^c=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#11119, #359161, #360231, … |
| maximal normal subgroup of | C2×#11119, #359161, #360231, … |
| maximal quotient of | #116438, #116475, #116479, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4107);