Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32706 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C2×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C25 | D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
492 conjugacy classes of subgroups; only the 109 normal subgroups are shown.
This group has 492 conjugacy classes of subgroups — too many to draw. Only its 109 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) = 64 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,6)(4,7)(5,8)(9,10)(11,13)(12,14)(15,16)(17,18,22,21)(19,23,26,29)(20,24,28,30)(25,31,32,27);(1,3)(2,5)(4,8)(6,7)(9,11)(10,13)(12,15)(14,16)(17,19,22,26)(18,23,21,29)(20,27,28,31)(24,25,30,32);(1,3)(2,6)(4,8)(5,7)(10,14)(13,16)(17,20,22,28)(18,24,21,30)(19,25,26,32)(23,31,29,27);(3,8)(5,6)(11,15)(13,16)Polycyclic presentation:
2^8|a^2=h,b^2=h,c^2=h,g^2=h,b^a=be,c^a=cf,c^b=cg,d^b=def,g^b=gh,d^c=de,g^c=gh40 conjugacy classes — view the full character table.
| maximal subgroup of | #122994, #123004, C2×#32711, … |
| maximal normal subgroup of | #122994, #123004, C2×#32711, … |
| maximal quotient of | #391192, #391193, #391233, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 25699);