Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32932 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C2×#127 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #376 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
359 conjugacy classes of subgroups; only the 119 normal subgroups are shown.
This group has 359 conjugacy classes of subgroups — too many to draw. Only its 119 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)(9,10)(11,13)(12,14)(15,16)(19,25)(20,27)(22,28)(24,30)(26,31)(29,32);(2,6)(7,8)(9,11,12,15)(10,13,14,16)(17,18,21,23)(19,22,26,29)(20,24,27,30)(25,28,31,32);(1,3,5,4)(2,7,6,8)(17,19,20,25,21,26,27,31)(18,22,24,28,23,29,30,32);(1,4,5,3)(2,8,6,7)(10,14)(13,16)(18,23)(22,29)(24,30)(28,32)Polycyclic presentation:
2^8|b^2=gh,c^2=ef,d^2=e,f^2=h,b^a=be,c^a=cf,d^a=dg,f^a=fh,d^b=dh52 conjugacy classes — view the full character table.
| maximal subgroup of | #125092, #125269, #125409, … |
| maximal normal subgroup of | #125092, #125269, #125409, … |
| maximal quotient of | #392543, #393184, #393188, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26272);