Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23918 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C22×D4 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C43 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
400 conjugacy classes of subgroups; only the 129 normal subgroups are shown.
This group has 400 conjugacy classes of subgroups — too many to draw. Only its 129 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) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,7)(4,5)(6,8)(17,18)(19,24)(20,23)(21,22);(2,5)(6,7)(9,10,12,14)(11,15,16,13)(17,19,20,21)(18,22,23,24);(9,11,12,16)(10,13,14,15)(18,22)(19,21)(23,24);(1,3,4,8)(2,6,5,7)(18,23)(22,24)Polycyclic presentation:
2^8|b^2=g,c^2=gh,d^2=eh,f^2=h,b^a=be,c^a=cf,d^a=de,f^a=fh,c^b=cg,f^c=fh52 conjugacy classes — view the full character table.
| maximal subgroup of | #101487, #101495, #166233, … |
| maximal normal subgroup of | #101487, #101495, #166233, … |
| maximal quotient of | #326559, #326568, #326577, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 22250);