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 | |
434 conjugacy classes of subgroups; only the 117 normal subgroups are shown.
This group has 434 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:
(1,2,6,10,7,11,5,20)(3,8,15,23,14,24,13,29)(4,16,19,9,18,26,21,25)(12,27,28,22,17,31,30,32);(1,3)(2,8)(4,17,18,12)(5,13)(6,15)(7,14)(9,22,25,32)(10,23)(11,24)(16,27,26,31)(19,30,21,28)(20,29);(1,4)(2,9)(3,12)(5,21)(6,19)(7,18)(8,22)(10,26)(11,25)(13,30)(14,17)(15,28)(16,20)(23,31)(24,32)(27,29)Polycyclic presentation:
2^8|a^2=g,b^2=d,e^2=h,g^2=h,b^a=bd,c^a=ce,e^a=eh,f^a=fh,c^b=cf,f^b=fh,d^c=dh,e^c=eh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#11094, #359071, #359792, … |
| maximal normal subgroup of | C2×#11094, #359071, #359792, … |
| maximal quotient of | #115979, #116002, #116003, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4082);