Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D4×D8 |
| automorphism | Aut(G) = C24×D4×#120 (order 4096) · Inn(G) = C22×D4 · Out(G) = C24×D4 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22×D4 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| socle | C22 | C2×C4×D4 |
| prime radical | C2×#135 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C4⋊C8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
376 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,6)(4,8)(5,7)(9,10,13,17,14,11,18,16)(12,19,22,20,23,15,21,24);(1,3,5,4)(2,6,7,8)(9,11)(10,14)(12,20)(13,16)(15,22)(17,18)(19,21)(23,24);(1,4,5,3)(2,6,7,8);(9,12)(10,15)(11,19)(13,22)(14,23)(16,20)(17,24)(18,21)Polycyclic presentation:
2^8|a^2=g,b^2=f,c^2=f,e^2=h,g^2=h,b^a=be,c^a=cf,d^a=dh,e^a=eh,d^b=de,e^b=eh,e^d=eh70 conjugacy classes — view the full character table.
| maximal subgroup of | #221029, #221031, #221035, … |
| maximal normal subgroup of | #221029, #221031, #221035, … |
| maximal quotient of | #226494, #226495, #226496, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 15771);