Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33605 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×D8 |
| commutator, Frattini, cosocle | C2×C8 | C24 |
| prime radical | #3147 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
311 conjugacy classes of subgroups; only the 123 normal subgroups are shown.
This group has 311 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)(3,6)(4,5)(7,8)(9,10)(11,21)(12,17)(13,19)(14,18)(15,23)(16,20)(22,24);(2,5)(6,8)(9,10,14,19,15,17,24,11,16,20,21,22,12,23,13,18);(1,3,4,7)(2,6,5,8)(9,11,13,17,12,19,21,10,16,18,24,23,15,22,14,20);(9,12,16,15)(10,17,20,23)(11,22,18,19)(13,21,24,14)Polycyclic presentation:
2^8|b^2=f,c^2=e,d^2=h,e^2=gh,f^2=g,g^2=h,b^a=be,c^a=cf,e^a=eg,f^a=fgh,g^a=gh,d^b=dh,d^c=dh52 conjugacy classes — view the full character table.
| maximal subgroup of | #137513, #137514, #137516, … |
| maximal normal subgroup of | #137513, #137514, #137516, … |
| maximal quotient of | #494087, #494914, #495121, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26695);