Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3264 |
| automorphism | Aut(G) = C28×D4 (order 2048) · Inn(G) = C22×D4 · Out(G) = C26 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
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 | #3002 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C22⋊C8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
529 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2)(3,4)(5,6,8,13,9,14,18,12)(7,11,16,19,17,20,15,10);(5,7,8,16,9,17,18,15)(6,10,13,11,14,19,12,20);(2,4);(5,7)(6,11)(8,16)(9,17)(10,12)(13,19)(14,20)(15,18)Polycyclic presentation:
2^8|a^2=g,b^2=g,e^2=h,g^2=h,b^a=be,c^a=cf,e^a=eh,d^b=de,e^b=eh,e^d=eh58 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#22886, #6381454, #6381456, … |
| maximal normal subgroup of | C2×#22886, #6381454, #6381456, … |
| maximal quotient of | #227154, #227155, #227156, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 15874);