Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12093 |
| automorphism | Aut(G) = C25×B(4,𝔽2) (order 2048) · Inn(G) = B(4,𝔽2) · Out(G) = C25 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | B(4,𝔽2) |
| commutator, Frattini, cosocle | C2×C22⋊C4 | C23 |
| prime radical | #392 | C22 |
| LCS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23 ◁ C2×C22⋊C4 ◁ G | |
| LECS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
206 conjugacy classes of subgroups; only the 35 normal subgroups are shown.
This group has 206 conjugacy classes of subgroups — too many to draw. Only its 35 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,5,10)(3,8,15,24)(4,7,16,9)(6,11,13,22)(12,25,27,32)(14,19,30,23)(17,21,29,31)(18,20,28,26);(2,7)(3,12)(4,16)(8,21)(9,10)(11,20)(14,29)(15,27)(17,30)(18,28)(19,25)(22,26)(23,32)(24,31);(1,3,13,27)(2,8,22,32)(4,17,28,30)(5,15,6,12)(7,19,26,31)(9,23,20,21)(10,24,11,25)(14,16,29,18)Polycyclic presentation:
2^8|a^2=f,c^2=fh,d^2=f,b^a=bd,c^a=ce,d^a=df,g^a=gh,c^b=ch,d^b=df,e^b=egh,d^c=dg,e^d=eh25 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#12231, #484414, #484416, … |
| maximal normal subgroup of | C2×#12231, #484414, #484416, … |
| maximal quotient of | #125905, #125936, #125974, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5219);