Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33557 |
| automorphism | Aut(G) = #72256403 (order 2048) · Inn(G) = C2×B(4,𝔽2) · Out(G) = C24 |
| 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×D4 | C24 |
| LCS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ C22×D4 ◁ G | |
| LECS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
629 conjugacy classes of subgroups; only the 109 normal subgroups are shown.
This group has 629 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)(3,7)(4,8)(5,9)(6,11)(10,15)(12,14)(13,16);(1,3)(2,6)(4,11)(5,10)(7,8)(9,14)(12,13)(15,16);(1,3)(2,7)(4,12)(5,10)(6,8)(9,15)(11,13)(14,16);(3,10)(6,14)(7,15)(11,12)Polycyclic presentation:
2^8|b^a=be,c^a=cf,g^a=gh,d^b=dh,f^b=fgh,d^c=dh,e^c=eg,f^e=fh34 conjugacy classes — view the full character table.
| maximal subgroup of | #152572, #153120, #153143, … |
| maximal normal subgroup of | #152572, #153120, #153143, … |
| maximal quotient of | #484162, #484176, #484248, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26545);