Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32712 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C2×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | #1999 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
410 conjugacy classes of subgroups; only the 103 normal subgroups are shown.
This group has 410 conjugacy classes of subgroups — too many to draw. Only its 103 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:
(1,2,4,5)(3,6,7,8)(9,10)(11,13)(12,14)(15,16)(17,18)(19,25)(20,22)(21,24)(23,27)(26,31)(28,30)(29,32);(2,5)(3,7)(9,11)(10,13)(12,15)(14,16)(18,22)(19,26)(21,28)(23,31)(25,32)(27,29);(3,7)(6,8)(10,14)(13,16)(17,19)(18,23)(20,27)(21,29)(22,25)(24,31)(26,28)(30,32);(1,3,4,7)(2,6,5,8)(11,15)(13,16)(18,22)(19,27)(24,30)(26,29)Polycyclic presentation:
2^8|a^2=eh,d^2=eh,g^2=h,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=def,g^b=gh,d^c=de,g^c=gh34 conjugacy classes — view the full character table.
| maximal subgroup of | #123037, #123047, C2×#32769, … |
| maximal normal subgroup of | #123037, #123047, C2×#32769, … |
| maximal quotient of | #391195, #391197, #391199, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 25757);