Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32712 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | #2619 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
372 conjugacy classes of subgroups; only the 103 normal subgroups are shown.
This group has 372 conjugacy classes of subgroups — too many to draw. Only its 103 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)(3,5)(4,6)(7,8)(9,10,13,12)(11,15,16,14)(17,18,21,23)(19,22,27,30)(20,24,28,32)(25,29,26,31);(1,3)(2,5)(4,7)(6,8)(9,11,13,16)(10,14,12,15)(17,19)(18,22)(20,25)(21,27)(23,30)(24,29)(26,28)(31,32);(2,6)(5,8)(10,12)(14,15)(19,25)(20,28)(22,29)(24,32)(26,27)(30,31);(3,7)(5,8)(9,12,13,10)(11,15,16,14)(18,23)(22,30)(24,32)(29,31)Polycyclic presentation:
2^8|a^2=eh,b^2=e,d^2=e,g^2=h,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=def,g^b=gh,d^c=de,g^c=gh34 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#32782, #6631176, #6631183, … |
| maximal normal subgroup of | C2×#32782, #6631176, #6631183, … |
| maximal quotient of | #391240, #391244, #391296, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 25770);