Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #29748 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#121 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C2×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
310 conjugacy classes of subgroups; only the 115 normal subgroups are shown.
This group has 310 conjugacy classes of subgroups — too many to draw. Only its 115 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)(3,7)(4,8)(5,6)(9,15)(10,13)(11,16)(12,14)(17,18,20,25,22,26,21,24)(19,23,28,31,29,30,27,32);(2,6)(7,13)(8,14)(15,16)(17,19)(18,23)(20,28)(21,27)(22,29)(24,32)(25,31)(26,30);(1,3,5,10)(2,7,6,13)(4,11,12,9)(8,16,14,15)(18,24)(19,27)(20,21)(23,30)(25,26)(28,29);(1,4,5,12)(2,8,6,14)(3,9,10,11)(7,15,13,16)Polycyclic presentation:
2^8|a^2=f,c^2=e,d^2=e,f^2=h,g^2=h,b^a=be,c^a=cf,c^b=cg,g^b=gh,d^c=de,f^c=fh,g^c=gh40 conjugacy classes — view the full character table.
| maximal subgroup of | #138489, #138490, #272543, … |
| maximal normal subgroup of | #138489, #138490, #272543, … |
| maximal quotient of | #330735, #330737, #330740, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 22874);