Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11549 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#99 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C2×C4⋊C4 | C23 |
| prime radical | C2×#127 | C4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ G | |
| LECS | G ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
231 conjugacy classes of subgroups; only the 53 normal subgroups are shown.
This group has 231 conjugacy classes of subgroups — too many to draw. Only its 53 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:
(1,2,4,8,5,6,3,7)(9,10)(11,15)(12,16,20,14)(13,17)(18,23)(19,22,24,21);(2,6)(3,4)(10,14)(11,18)(12,20)(15,22)(16,17)(21,23);(1,3,5,4)(2,7,6,8)(9,11,13,18)(10,15,17,23)(12,19,20,24)(14,21,16,22)Polycyclic presentation:
2^8|a^2=e,c^2=gh,d^2=h,e^2=g,b^a=bd,d^a=df,f^a=fh,c^b=cgh,d^b=dh,e^b=efh,e^d=eh37 conjugacy classes — view the full character table.
| maximal subgroup of | #132205, #132207, #132209, … |
| maximal normal subgroup of | #132205, #132207, #132209, … |
| maximal quotient of | #125167, #125211, #125233, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4555);