Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #468 |
| automorphism | Aut(G) = C2×#72233548 (order 2048) · Inn(G) = C2×D4 · Out(G) = C2×#525 |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C8 | C2×D4 |
| commutator | C2×C4 | C22×C8 |
| Frattini, cosocle | C4×C8 | C23 |
| socle | C22 | C2×C4⋊C8 |
| prime radical | C23 | C4⋊C8 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
116 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 24 points:
(2,5)(3,7)(6,8)(9,10,11,15,12,16,17,22,13,14,18,20,19,21,23,24);(1,2,3,5,4,6,7,8);(2,6)(5,8)(10,14)(15,20)(16,21)(22,24)Polycyclic presentation:
2^8|a^2=e,b^2=d,d^2=f,e^2=g,g^2=h,b^a=bd,c^a=ch,d^a=df,c^b=cf64 conjugacy classes — view the full character table.
| maximal subgroup of | #135858, #135862, #135869, … |
| maximal normal subgroup of | #135858, #135862, #135869, … |
| maximal quotient of | #126677, #129118, #134684, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5005);