Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #29736 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C24 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
366 conjugacy classes of subgroups; only the 119 normal subgroups are shown.
This group has 366 conjugacy classes of subgroups — too many to draw. Only its 119 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 28 points:
(5,6,8,9)(7,10,11,12)(13,14,17,22,18,21,27,28)(15,19,23,26,20,16,24,25);(1,2)(3,4)(6,9)(10,12)(13,15)(14,19)(16,21)(17,23)(18,20)(22,26)(24,27)(25,28);(2,4)(7,11)(10,12)(13,16,18,19)(14,20,21,15)(17,26,27,25)(22,23,28,24);(5,7,8,11)(6,10,9,12)(14,21)(15,20)(22,28)(23,24)Polycyclic presentation:
2^8|a^2=ef,c^2=h,d^2=e,f^2=h,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=dh,d^c=de,f^c=fh40 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#29856, #6435540, #6435548, … |
| maximal normal subgroup of | C2×#29856, #6435540, #6435548, … |
| maximal quotient of | #330520, #330523, #330526, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 22844);