Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32929 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C22×D4 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C43 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
354 conjugacy classes of subgroups; only the 127 normal subgroups are shown.
This group has 354 conjugacy classes of subgroups — too many to draw. Only its 127 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) = 128 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,7)(4,8)(5,6)(9,10)(11,13)(12,14)(15,16)(17,18)(19,25)(20,28)(21,23)(22,26)(24,29)(27,31)(30,32);(2,6)(7,8)(9,11,12,15)(10,13,14,16);(1,3,5,4)(2,7,6,8)(17,19,20,26,21,27,29,32)(18,22,24,25,23,30,28,31);(1,4,5,3)(2,8,6,7)(10,14)(13,16)(18,23)(22,30)(24,28)(25,31)Polycyclic presentation:
2^8|b^2=gh,c^2=ef,d^2=e,f^2=h,b^a=be,c^a=cf,d^a=dg,f^a=fh76 conjugacy classes — view the full character table.
| maximal subgroup of | #6474768, #6474769, #6474772, … |
| maximal normal subgroup of | #6474768, #6474769, #6474772, … |
| maximal quotient of | #392532, #393179, #393267, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26267);