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) = C2×C4 |
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 | |
211 conjugacy classes of subgroups; only the 35 normal subgroups are shown.
This group has 211 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)(3,8,14,23)(4,7,15,9)(6,11,18,26)(12,24,27,32)(13,19,30,22)(16,21,28,31)(17,20,29,25);(2,7)(3,12)(4,15)(8,21)(9,10)(11,20)(13,28)(14,27)(16,30)(17,29)(19,24)(22,32)(23,31)(25,26);(1,3,5,14)(2,8,10,23)(4,16,15,28)(6,12,18,27)(7,19,9,22)(11,24,26,32)(13,29,30,17)(20,21,25,31)Polycyclic presentation:
2^8|a^2=f,c^2=f,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×#12230, #484414, #484415, … |
| maximal normal subgroup of | C2×#12230, #484414, #484415, … |
| maximal quotient of | #125905, #125935, #125972, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5218);