Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #7855 |
| automorphism | Aut(G) = #72303155 (order 4096) · Inn(G) = C2×C42 · Out(G) = C2×#379 |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator | C2×C4 | C2×C42 |
| Frattini, cosocle | C4×C8 | C23 |
| socle | C22 | C2×#97 |
| prime radical | C23 | #97 |
| LCS | G ▷ C2×C4 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
123 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,4,7,5,8,11,14)(3,6,10,13,9,12,15,16)(18,22,23,28)(19,25)(24,31,30,29)(26,32);(1,3)(2,6)(4,10)(5,9)(7,13)(8,12)(11,15)(14,16)(17,18,19,24,20,22,26,29,21,23,25,30,27,28,32,31);(3,9)(6,12)(10,15)(13,16)Polycyclic presentation:
2^8|a^2=d,b^2=e,d^2=f,e^2=g,g^2=h,b^a=bg,e^a=eh,c^b=cf,d^b=dh64 conjugacy classes — view the full character table.
| maximal subgroup of | #128231, #128232, #128233, … |
| maximal normal subgroup of | #128231, #128232, #128233, … |
| maximal quotient of | #126537, #126574, #128217, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4403);