Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #7855 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator | C2×C4 | C2×C42 |
| Frattini, cosocle | C2×C42 | C23 |
| socle | C22 | C2×#95 |
| prime radical | C24 | C4⋊C4 |
| LCS | G ▷ C2×C4 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
313 conjugacy classes of subgroups; only the 131 normal subgroups are shown.
This group has 313 conjugacy classes of subgroups — too many to draw. Only its 131 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2,3,6,4,5,7,8)(9,10)(11,18,20,24)(12,16)(13,22)(14,17)(15,19,23,21);(9,11,13,21,14,20,12,19)(10,15,16,18,17,23,22,24);(2,5)(6,8)Polycyclic presentation:
2^8|a^2=e,b^2=f,d^2=g,e^2=h,f^2=g,b^a=bd,c^a=ch,f^a=fg,e^b=eg64 conjugacy classes — view the full character table.
| maximal subgroup of | #93567, #93568, #105589, … |
| maximal normal subgroup of | #93567, #93568, #105589, … |
| maximal quotient of | #94864, #94866, #94871, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 849);