Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #7402 |
| automorphism | Aut(G) = C22×#501097 (order 2048) · Inn(G) = #353 · Out(G) = C22×D4 |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #353 |
| commutator | C2×C42 | C2×C4 |
| Frattini, cosocle | #391 | C22 |
| prime radical | C23 | #99 |
| LCS | G ▷ C2×C42 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23 ◁ C22×C4 ◁ #391 ◁ G | |
| LECS | G ▷ #391 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
97 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 2 generators on 24 points:
(2,5,6,7)(9,10,12,17,14,19,11,20)(13,16,21,15,22,18,23,24);(1,2)(3,7)(4,6)(5,8)(9,11,13,21,14,12,22,23)(10,15,16,17,19,24,18,20)Polycyclic presentation:
2^8|a^2=d,b^2=eg,c^2=fgh,d^2=h,e^2=h,b^a=bc,c^a=ce,e^a=eg,g^a=gh,c^b=cf,d^b=defg,d^c=dgh,e^d=eh28 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#7436, #150290, #150291, … |
| maximal normal subgroup of | C2×#7436, #150290, #150291, … |
| maximal quotient of | #93243, #93247, #93957, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 424);