Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11692 |
| 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×C22⋊C4 | C23 |
| prime radical | C25 | D4 |
| LCS | G ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C22×C22⋊C4 ◁ G | |
| LECS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
296 conjugacy classes of subgroups; only the 63 normal subgroups are shown.
This group has 296 conjugacy classes of subgroups — too many to draw. Only its 63 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) = 32 |
A minimal faithful permutation representation — 3 generators on 24 points:
(2,5,6,7)(9,10)(11,17)(12,20,22,15)(13,16)(14,19,23,21)(18,24);(1,2)(3,7)(4,6)(5,8)(9,11)(10,14,16,23)(12,21)(13,18)(15,24,20,17)(19,22);(11,18)(14,23)(17,24)(19,21)Polycyclic presentation:
2^8|a^2=e,b^2=f,d^2=gh,b^a=bd,d^a=df,f^a=fh,c^b=ch,d^b=dg,e^b=efgh,e^d=eh40 conjugacy classes — view the full character table.
| maximal subgroup of | #129805, #129806, #129821, … |
| maximal normal subgroup of | #129805, #129806, #129821, … |
| maximal quotient of | #124090, #124122, #124195, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4704);