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) = C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #493 |
| commutator, Frattini, cosocle | C4×C8 | C23 |
| prime radical | #493 | C22 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ C82 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
130 conjugacy classes of subgroups; only the 47 normal subgroups are shown.
This group has 130 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) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,5,8)(3,9,10,16)(4,12,13,7)(6,14,15,11)(18,21)(19,27)(20,23)(24,29)(25,26)(28,31)(30,32);(1,3,4,11,5,10,13,14)(2,6,7,9,8,15,12,16)(17,18,23,26,22,25,20,21)(19,28,31,27,30,24,29,32);(3,10)(6,15)(9,16)(11,14)(17,19,18,24,23,31,26,32,22,30,25,28,20,29,21,27)Polycyclic presentation:
2^8|a^2=fh,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×#12536, #491528, #491529, … |
| maximal normal subgroup of | C2×#12536, #491528, #491529, … |
| maximal quotient of | #135788, #141710, #141713, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5524);