Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #447 |
| automorphism | Aut(G) = C25×C4×D4 (order 1024) · Inn(G) = C2×D4 · Out(G) = C24×C4 |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C8 | C2×D4 |
| commutator | C2×C4 | C22×C8 |
| Frattini, cosocle | C22×C8 | C23 |
| socle | C22 | C8×D4 |
| prime radical | C2×#41 | C2×C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ C22×C16 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
192 conjugacy classes of subgroups; only the 83 normal subgroups are shown.
This group has 192 conjugacy classes of subgroups — too many to draw. Only its 83 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)(5,6,11,17,12,16,29,34,9,15,25,18,10,26,28,35)(7,8,21,24,22,23,14,32,19,13,30,31,20,27,33,36);(2,4)(5,7,11,21,12,22,29,14,9,19,25,30,10,20,28,33)(6,13,17,31,16,27,34,36,15,8,18,24,26,23,35,32);(5,8)(6,14)(7,18)(9,13)(10,27)(11,24)(12,23)(15,33)(16,30)(17,19)(20,34)(21,26)(22,35)(25,31)(28,36)(29,32)Polycyclic presentation:
2^8|a^2=f,b^2=f,e^2=h,f^2=g,g^2=h,b^a=bd,c^a=ce,e^a=eh,e^c=eh76 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#12917, #494229, #494230, … |
| maximal normal subgroup of | C2×#12917, #494229, #494230, … |
| maximal quotient of | #135819, #135831, #135836, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5905);