Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #29742 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23 |
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 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
351 conjugacy classes of subgroups; only the 127 normal subgroups are shown.
This group has 351 conjugacy classes of subgroups — too many to draw. Only its 127 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 — 4 generators on 32 points:
(1,2)(3,6)(4,5)(7,8)(9,10,12,14)(11,15,16,13)(17,18,21,25)(19,27,30,23)(20,24,22,26)(28,32,29,31);(2,5)(6,8)(17,19,22,29,21,30,20,28)(18,23,26,32,25,27,24,31);(1,3)(2,6)(4,7)(5,8)(9,11,12,16)(10,13,14,15)(19,28)(20,22)(23,31)(24,26)(27,32)(29,30);(3,7)(6,8)(17,20,21,22)(18,24,25,26)(19,29,30,28)(23,32,27,31)Polycyclic presentation:
2^8|a^2=fh,b^2=g,c^2=f,d^2=h,g^2=h,b^a=be,c^a=cf,c^b=cg,d^b=dh,d^c=de,g^c=gh40 conjugacy classes — view the full character table.
| maximal subgroup of | #144457, #144461, #144487, … |
| maximal normal subgroup of | #144457, #144461, #144487, … |
| maximal quotient of | #380805, #380806, #380969, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 23285);