Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #1404 |
| automorphism | Aut(G) = #72286229 (order 2048) · Inn(G) = AΣL(1,𝔽16) · Out(G) = C22×D4 |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | AΣL(1,𝔽16) |
| commutator | C2×Q8 | C42 |
| Frattini, prime radical, cosocle | #532 | C22 |
| socle | C2 | #1390 |
| LCS | G ▷ C2×Q8 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C2×C4 ◁ C22×C4 ◁ #532 ◁ G | |
| LECS | G ▷ #532 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
137 conjugacy classes of subgroups; only the 33 normal subgroups are shown.
This group has 137 conjugacy classes of subgroups — too many to draw. Only its 33 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 — 2 generators on 32 points:
(1,2,4,9)(3,13,15,7)(5,10,20,27)(6,12,22,29)(8,23,11,21)(14,30,17,28)(16,31,19,24)(18,26,32,25);(1,3,5,16,6,18,23,17)(2,7,10,24,12,25,11,30)(4,19,20,32,22,14,21,15)(8,26,9,28,27,13,29,31)Polycyclic presentation:
2^8|a^2=d,b^2=e,c^2=h,e^2=h,f^2=h,b^a=bc,c^a=cf,f^a=fg,g^a=gh,c^b=ch,d^b=df,f^b=fh,g^b=gh,d^c=dgh,f^c=fh,f^d=fh34 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#7490, #151806, #151822, … |
| maximal normal subgroup of | C2×#7490, #151806, #151822, … |
| maximal quotient of | #94089, #94133, #94144, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 478);