Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2008 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #392 |
| 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 | |
426 conjugacy classes of subgroups; only the 117 normal subgroups are shown.
This group has 426 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:
(1,2,6,10,7,11,5,21)(3,12,16,23,15,24,14,8)(4,9,19,26,20,28,18,27)(13,29,31,22,30,25,17,32);(1,3,5,14,7,15,6,16)(2,8,10,12,11,23,21,24)(4,17,18,30,20,31,19,13)(9,25,26,32,28,29,27,22);(1,4)(2,9)(3,13)(5,18)(6,19)(7,20)(8,22)(10,26)(11,28)(12,25)(14,17)(15,30)(16,31)(21,27)(23,32)(24,29)Polycyclic presentation:
2^8|a^2=g,b^2=d,d^2=h,f^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dh,f^a=fh,c^b=cf,e^b=eh,f^b=fh,f^c=fh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#11118, #359071, #360230, … |
| maximal normal subgroup of | C2×#11118, #359071, #360230, … |
| maximal quotient of | #116437, #116476, #116479, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4106);