Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #1404 |
| automorphism | Aut(G) = #72249492 (order 2048) · Inn(G) = AΣL(1,𝔽16) · Out(G) = C22×D4 |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | AΣL(1,𝔽16) |
| commutator | C2×Q8 | C42 |
| Frattini, cosocle | C2×C4⋊Q8 | C22 |
| prime radical | C23 | #99 |
| LCS | G ▷ C2×Q8 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23 ◁ C22×C4 ◁ C2×C4⋊Q8 ◁ G | |
| LECS | G ▷ C2×C4⋊Q8 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
138 conjugacy classes of subgroups; only the 35 normal subgroups are shown.
This group has 138 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) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 2 generators on 20 points:
(5,6,9,13,10,7,8,16)(11,14,18,15,20,17,19,12);(1,2,3,4)(5,7,10,6)(8,12,9,15)(11,14,20,17)(13,19,16,18)Polycyclic presentation:
2^8|a^2=d,b^2=e,c^2=h,d^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×#7491, #151808, #151809, … |
| maximal normal subgroup of | C2×#7491, #151808, #151809, … |
| maximal quotient of | #94088, #94133, #94145, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 479);