Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7654 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | C2×C22⋊C4 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C2×C42 | C23 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C43 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C23 ▷ C1 | |
216 conjugacy classes of subgroups; only the 77 normal subgroups are shown.
This group has 216 conjugacy classes of subgroups — too many to draw. Only its 77 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) = 256 |
A minimal faithful permutation representation — 3 generators on 24 points:
(2,6,7,8)(9,10,12,14,13,16,11,15)(17,18,21,23,22,19,20,24);(1,2)(3,6)(4,8)(5,7)(10,14)(11,12)(15,16)(17,19,22,18)(20,23,21,24);(1,3,5,4)(2,6,7,8)(9,11,13,12)(10,15,16,14)(18,19)(23,24)Polycyclic presentation:
2^8|a^2=e,b^2=f,c^2=fg,d^2=g,e^2=h,b^a=bd,c^a=cf,d^a=df,c^b=ch,d^b=dg,e^b=efg58 conjugacy classes — view the full character table.
| maximal subgroup of | #97185, #97186, #97191, … |
| maximal normal subgroup of | #97185, #97186, #97191, … |
| maximal quotient of | #96285, #96753, #96929, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 668);