Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30396 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
320 conjugacy classes of subgroups; only the 109 normal subgroups are shown.
This group has 320 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) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,4,5)(3,7,8,6)(9,10)(11,15)(12,14)(13,16);(2,5)(3,8)(9,11)(10,13)(12,16)(14,15)(17,18,22,25)(19,26,28,31)(20,24,21,29)(23,30,32,27);(10,14)(13,15)(17,19,22,28)(18,23,25,32)(20,27,21,30)(24,31,29,26);(1,3,4,8)(2,6,5,7)(11,16)(13,15)(17,20,22,21)(18,24,25,29)(19,27,28,30)(23,31,32,26)Polycyclic presentation:
2^8|a^2=ef,b^2=h,c^2=h,d^2=efh,g^2=h,b^a=be,c^a=cf,d^a=def,c^b=cg,d^b=de,g^b=gh,g^c=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | #123060, #123070, #6647793, … |
| maximal normal subgroup of | #123060, #123070, #6647793, … |
| maximal quotient of | #384820, #385816, #386652, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 25537);