Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11796 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
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 | C22×D4 | C2×C4 |
| LCS | G ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ G | |
| LECS | G ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
208 conjugacy classes of subgroups; only the 51 normal subgroups are shown.
This group has 208 conjugacy classes of subgroups — too many to draw. Only its 51 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:
(2,6)(3,4)(7,8)(9,10,12,16,13,14,19,21)(11,15,17,22,18,20,23,24);(1,2)(3,6)(4,8)(5,7)(10,14)(15,20);(1,3,5,4)(2,6,7,8)(9,11,13,18)(10,15,14,20)(12,17,19,23)(16,22,21,24)Polycyclic presentation:
2^8|a^2=e,c^2=gh,d^2=g,e^2=h,b^a=bd,c^a=cg,d^a=df,f^a=fh,d^b=dg,e^b=efg,e^d=eh34 conjugacy classes — view the full character table.
| maximal subgroup of | #132205, #132206, #132209, … |
| maximal normal subgroup of | #132205, #132206, #132209, … |
| maximal quotient of | #124568, #124579, #124593, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4817);