Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12148 |
| 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 | C25 | D4 |
| LCS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23 ◁ C2×C22⋊C4 ◁ G | |
| LECS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
234 conjugacy classes of subgroups; only the 35 normal subgroups are shown.
This group has 234 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:
(1,2,6,10)(3,12,16,23)(4,17,19,27)(5,20,18,9)(7,11,22,26)(8,21,24,14)(13,25,30,32)(15,28,29,31);(1,3,7,15)(2,8,11,25)(4,13,18,14)(5,21,19,30)(6,16,22,29)(9,23,17,31)(10,24,26,32)(12,27,28,20);(1,4,6,19)(2,9,26,27)(3,13,16,30)(5,7,18,22)(8,23,32,28)(10,20,11,17)(12,25,31,24)(14,29,21,15)Polycyclic presentation:
2^8|a^2=f,b^2=h,c^2=fg,d^2=f,b^a=bd,c^a=ce,d^a=df,g^a=gh,d^b=df,e^b=egh,d^c=dg,e^c=eh,e^d=eh28 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#12264, #484822, #484828, … |
| maximal normal subgroup of | C2×#12264, #484822, #484828, … |
| maximal quotient of | #125922, #125952, #126004, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5252);