Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8304 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×C22⋊C4 |
| commutator | C24 | C22×C4 |
| Frattini, cosocle | C23×C4 | C23 |
| prime radical | C25 | C2×C4 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
415 conjugacy classes of subgroups; only the 79 normal subgroups are shown.
This group has 415 conjugacy classes of subgroups — too many to draw. Only its 79 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)(11,15,16,13)(12,14)(17,18,20,24,21,23,19,22);(2,5)(6,8)(17,19,21,20)(18,22,23,24);(1,3,4,7)(2,6,5,8)(9,11)(10,13)(12,16)(14,15)(18,23)(19,20)Polycyclic presentation:
2^8|a^2=f,b^2=h,c^2=d,f^2=h,b^a=bd,c^a=ce,e^a=eg,c^b=ch,f^c=fg52 conjugacy classes — view the full character table.
| maximal subgroup of | #97446, #97447, #97451, … |
| maximal normal subgroup of | #97446, #97447, #97451, … |
| maximal quotient of | #99117, #99118, #99119, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 1692);