Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12525 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C8 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #493 |
| commutator, Frattini, cosocle | C4×C8 | C23 |
| prime radical | #2246 | C2 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ C82 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
153 conjugacy classes of subgroups; only the 47 normal subgroups are shown.
This group has 153 conjugacy classes of subgroups — too many to draw. Only its 47 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 — 3 generators on 24 points:
(2,5)(3,7)(6,8)(10,13)(11,19)(12,15)(16,21)(17,18)(20,23)(22,24);(1,2,3,5,4,6,7,8)(9,10,15,18,14,17,12,13)(11,20,23,19,22,16,21,24);(2,6)(5,8)(9,11,10,16,15,23,18,24,14,22,17,20,12,21,13,19)Polycyclic presentation:
2^8|b^2=dh,c^2=eg,d^2=f,e^2=gh,g^2=h,b^a=bd,c^a=ce,d^a=df,e^a=eg,g^a=gh,c^b=cf46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#12527, #491510, #491511, … |
| maximal normal subgroup of | C2×#12527, #491510, #491511, … |
| maximal quotient of | #135788, #141712, #141713, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5515);