Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33568 |
| automorphism | Aut(G) = #72342352 (order 4096) · Inn(G) = C2×B(4,𝔽2) · Out(G) = C25 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C2×B(4,𝔽2) |
| commutator, Frattini, cosocle | C2×Q8 | C24 |
| LCS | G ▷ C2×Q8 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×#41 ◁ G | |
| LECS | G ▷ C2×Q8 ▷ C22 ▷ C2 ▷ C1 | |
606 conjugacy classes of subgroups; only the 109 normal subgroups are shown.
This group has 606 conjugacy classes of subgroups — too many to draw. Only its 109 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 16 |
A minimal faithful permutation representation — 4 generators on 16 points:
(1,2)(3,9)(4,7)(5,8)(6,11)(10,14)(12,16)(13,15);(2,7)(3,4)(5,14)(8,15)(9,11)(10,16);(2,8)(5,14)(7,15)(9,16)(10,11)(12,13);(1,3,6,4)(2,9,11,7)(5,12,14,13)(8,16,10,15)Polycyclic presentation:
2^8|d^2=h,e^2=h,f^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fh,g^a=gh,d^b=dh,e^b=eh,f^b=fgh,e^c=eg,f^c=fh,f^e=fh34 conjugacy classes — view the full character table.
| maximal subgroup of | #152564, #152570, #153127, … |
| maximal normal subgroup of | #152564, #152570, #153127, … |
| maximal quotient of | #484178, #484183, #484264, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26558);