Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18132 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
691 conjugacy classes of subgroups; only the 115 normal subgroups are shown.
This group has 691 conjugacy classes of subgroups — too many to draw. Only its 115 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) = 32 |
A minimal faithful permutation representation — 4 generators on 16 points:
(2,6)(3,4)(7,8)(13,16);(1,2)(3,6)(4,8)(5,7)(9,10)(11,13)(12,14)(15,16);(9,11,12,15)(10,13,14,16);(1,3,5,4)(2,6,7,8)Polycyclic presentation:
2^8|c^2=h,d^2=g,e^2=g,b^a=be,c^a=cf,d^a=dg,e^a=eg,e^b=eg,f^b=fh,e^c=eh37 conjugacy classes — view the full character table.
| maximal subgroup of | #142476, #142477, #142492, … |
| maximal normal subgroup of | #142476, #142477, #142492, … |
| maximal quotient of | #165886, #165897, #165925, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 11133);