Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1812 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #117 |
| commutator | C2×C4 | C2×C42 |
| Frattini, cosocle | C23×C4 | C23 |
| socle | C22 | #382 |
| prime radical | C24 | #41 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
321 conjugacy classes of subgroups; only the 135 normal subgroups are shown.
This group has 321 conjugacy classes of subgroups — too many to draw. Only its 135 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(1,2)(3,4)(5,6)(7,17,20,12)(8,22,24,30)(9,15)(10,16)(11,14)(13,23,29,25)(18,32)(19,28,21,27)(26,33)(31,34)(35,36);(5,7,10,21,11,20,9,19)(6,12,16,28,14,17,15,27)(8,18,25,34,24,35,23,33)(13,26,30,32,29,31,22,36);(2,4)(5,8)(6,13,14,29)(7,18)(9,23)(10,25)(11,24)(12,26,17,31)(15,22,16,30)(19,33)(20,35)(21,34)(27,36,28,32)Polycyclic presentation:
2^8|a^2=f,b^2=g,c^2=d,e^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dh,f^b=fh,f^c=fh64 conjugacy classes — view the full character table.
| maximal subgroup of | #93548, #134475, #134476, … |
| maximal normal subgroup of | #93548, #134475, #134476, … |
| maximal quotient of | #108500, #108501, #108502, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 3805);