Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32935 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C2×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C2×#127 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C43 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
399 conjugacy classes of subgroups; only the 123 normal subgroups are shown.
This group has 399 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) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,7)(4,8)(5,6)(9,10)(11,13)(12,14)(15,16)(18,21)(19,23)(22,24);(2,6)(7,8)(9,11,12,15)(10,13,14,16)(18,22)(21,24);(1,3,5,4)(2,7,6,8)(17,18,19,21,20,22,23,24);(1,4,5,3)(2,8,6,7)(10,14)(13,16)Polycyclic presentation:
2^8|b^2=g,c^2=ef,d^2=e,f^2=h,b^a=be,c^a=cf,d^a=dg,f^a=fh,c^b=ch52 conjugacy classes — view the full character table.
| maximal subgroup of | #295083, #295084, #295113, … |
| maximal normal subgroup of | #295083, #295084, #295113, … |
| maximal quotient of | #392538, #393191, #393194, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26268);