Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33638 |
| automorphism | Aut(G) = #72311826 (order 2048) · Inn(G) = C2×#476 · Out(G) = C24 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C2×#476 |
| commutator, Frattini, cosocle | C2×C8 | C24 |
| prime radical | C2×#135 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ #405 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
257 conjugacy classes of subgroups; only the 117 normal subgroups are shown.
This group has 257 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:
(2,8)(3,13,15,31)(4,6)(5,17)(9,23,24,27)(10,12)(14,18,20,30)(16,19)(21,25,29,32)(22,28);(1,2,4,8,7,10,6,12)(3,9,14,21,15,24,20,29)(5,11,16,22,19,26,17,28)(13,25,30,23,31,32,18,27);(1,3)(2,9)(4,14)(5,18)(6,20)(7,15)(8,21)(10,24)(11,27)(12,29)(13,16)(17,31)(19,30)(22,25)(23,26)(28,32);(2,10)(8,12)(9,24)(11,26)(21,29)(22,28)(23,27)(25,32)Polycyclic presentation:
2^8|a^2=eg,b^2=e,e^2=h,f^2=g,g^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fg,g^a=gh,d^b=dh,f^c=fgh,g^c=gh40 conjugacy classes — view the full character table.
| maximal subgroup of | #500471, #500472, #500473, … |
| maximal normal subgroup of | #500471, #500472, #500473, … |
| maximal quotient of | #493789, #494324, #494325, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26626);