Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | Aut(G) = C2×#120×#525 (order 4096) · Inn(G) = C23 · Out(G) = C23×#525 |
| homology | Abelianization (H1) = C23×C8 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4×C8 | C23 |
| commutator, socle | C22 | C23×C8 |
| Frattini, cosocle | C2×C8 | C24 |
| prime radical | C2×#41 | C8 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C4×C8 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
279 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,6)(4,5)(7,8)(9,10,12,16,13,11,18,19,14,17,21,24,22,20,15,23);(2,5)(6,8);(9,11,12,19,13,17,18,24,14,20,21,23,22,10,15,16);(1,3,4,7)(2,6,5,8)Polycyclic presentation:
2^8|a^2=f,c^2=f,d^2=e,f^2=g,g^2=h,b^a=be,c^a=ch112 conjugacy classes — view the full character table.
| maximal subgroup of | #128642, #128643, #128648, … |
| maximal normal subgroup of | #128642, #128643, #128648, … |
| maximal quotient of | #493188, #493189, C4×#2169, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26467);