Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2121 |
| automorphism | Aut(G) = #72303342 (order 4096) · Inn(G) = C2×#99 · Out(G) = C2×#120 |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | C2×#99 |
| commutator | C23 | C22×C8 |
| Frattini, cosocle | C2×M16 | C23 |
| socle | C2 | C2×#323 |
| prime radical | C22×#41 | C4 |
| LCS | G ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C22×M16 ◁ G | |
| LECS | G ▷ C2×M16 ▷ C2×C4 ▷ C2 ▷ C1 | |
252 conjugacy classes of subgroups; only the 69 normal subgroups are shown.
This group has 252 conjugacy classes of subgroups — too many to draw. Only its 69 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 — 3 generators on 32 points:
(1,2,4,9,5,10,16,24,6,8,15,21,17,22,28,31)(3,11,13,23,14,25,27,32,12,7,18,19,26,20,29,30);(1,3)(2,7)(4,13)(5,14)(6,12)(8,11)(9,23)(10,20)(15,18)(16,27)(17,26)(19,21)(22,25)(24,32)(28,29)(30,31);(3,12)(7,11)(13,18)(14,26)(19,23)(20,25)(27,29)(30,32)Polycyclic presentation:
2^8|a^2=e,e^2=g,g^2=h,b^a=bd,d^a=df,f^a=fh,c^b=ch,e^b=ef,e^d=eh46 conjugacy classes — view the full character table.
| maximal subgroup of | #129228, #129239, #129305, … |
| maximal normal subgroup of | #129228, #129239, #129305, … |
| maximal quotient of | #122225, #122226, #122227, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4424);