Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2230 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #476 |
| commutator, Frattini, cosocle | C2×C8 | C24 |
| socle | C2 | C2×#476 |
| prime radical | C2×#135 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
262 conjugacy classes of subgroups; only the 117 normal subgroups are shown.
This group has 262 conjugacy classes of subgroups — too many to draw. Only its 117 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) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,15)(4,19,22,32)(5,11)(6,13)(7,23)(8,14)(9,17)(10,16,26,31)(12,18)(20,29,21,27)(24,28,30,25);(1,3,6,17,8,18,5,7)(2,9,11,15,14,23,13,12)(4,16,21,30,22,31,20,24)(10,19,25,29,26,32,28,27);(1,4)(2,10)(3,16)(5,20)(6,21)(7,24)(8,22)(9,19)(11,25)(12,27)(13,28)(14,26)(15,29)(17,30)(18,31)(23,32);(1,5,8,6)(2,11,14,13)(3,7,18,17)(4,20,22,21)(9,15,23,12)(10,25,26,28)(16,24,31,30)(19,29,32,27)Polycyclic presentation:
2^8|a^2=eg,b^2=e,d^2=h,e^2=h,f^2=g,g^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fg,g^a=gh,f^c=fgh,g^c=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | #136858, #136860, #136862, … |
| maximal normal subgroup of | #136858, #136860, #136862, … |
| maximal quotient of | #493788, #494323, #494799, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26625);