Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8101 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#95 |
| commutator | C23 | C2×C42 |
| Frattini, cosocle | C23×C4 | C23 |
| prime radical | C25 | C2×C4 |
| LCS | G ▷ C23 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
471 conjugacy classes of subgroups; only the 123 normal subgroups are shown.
This group has 471 conjugacy classes of subgroups — too many to draw. Only its 123 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:
(1,2)(3,7)(4,6)(5,8)(9,10)(11,17,18,14)(12,16)(13,15)(19,23,24,21)(20,22);(1,3)(2,5,6,7)(4,8)(9,11,12,19,13,18,20,24)(10,14,16,21,15,17,22,23);(3,8)(5,7)Polycyclic presentation:
2^8|a^2=e,b^2=f,f^2=g,b^a=bd,d^a=dg,f^a=fh,c^b=ch,d^b=dh,e^b=eg52 conjugacy classes — view the full character table.
| maximal subgroup of | #106656, #106657, #106670, … |
| maximal normal subgroup of | #106656, #106657, #106670, … |
| maximal quotient of | #95112, #95117, #95124, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 1102);