Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(5,𝔽2) |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C23×C16 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C2×C8 | C24 |
| commutator | C2 | C23×C16 |
| socle | C22 | C23×C8 |
| prime radical | C22×D4 | C8 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
331 conjugacy classes of subgroups; only the 227 normal subgroups are shown.
This group has 331 conjugacy classes of subgroups — too many to draw. Only its 227 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,7)(4,6)(5,8)(9,10,11,14,12,15,17,20,13,16,18,21,19,22,23,24);(1,3)(2,5)(4,8)(6,7);(3,8)(5,7);(2,6)(5,7)Polycyclic presentation:
2^8|a^2=f,f^2=g,g^2=h,b^a=be,d^a=de,c^b=ce136 conjugacy classes — view the full character table.
| maximal subgroup of | #135853, #135934, #136767, … |
| maximal normal subgroup of | #135853, #135934, #136767, … |
| maximal quotient of | #493263, #493276, #493304, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26474);