Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30396 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
610 conjugacy classes of subgroups; only the 125 normal subgroups are shown.
This group has 610 conjugacy classes of subgroups — too many to draw. Only its 125 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:
(1,2,3,4)(9,10,13,16,14,17,12,22)(11,18,20,24,21,15,19,23);(5,6)(7,8)(9,10)(11,18)(12,16)(13,22)(14,17)(15,21)(19,24)(20,23);(2,4)(6,8)(9,11,14,21)(10,15,17,18)(12,19,13,20)(16,23,22,24);(2,4)(9,12)(11,19)(13,14)(16,22)(20,21)(23,24)Polycyclic presentation:
2^8|a^2=efh,c^2=h,e^2=h,b^a=be,c^a=cf,d^a=def,c^b=cg,d^b=de,e^b=eh,e^d=eh46 conjugacy classes — view the full character table.
| maximal subgroup of | #123053, #123063, #6634512, … |
| maximal normal subgroup of | #123053, #123063, #6634512, … |
| maximal quotient of | #343021, #343022, #343023, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 24047);