Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23897 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #2403 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
836 conjugacy classes of subgroups; only the 141 normal subgroups are shown.
This group has 836 conjugacy classes of subgroups — too many to draw. Only its 141 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)(3,11)(4,8)(5,17)(6,10)(7,14)(9,20)(12,28)(13,25)(15,26)(16,24)(18,21)(19,23)(22,30)(27,29)(31,32);(7,21)(11,25)(22,32)(27,28);(1,3)(2,7)(4,12)(5,18)(6,13)(8,22)(9,25)(10,21)(11,17)(14,20)(15,31)(16,29)(19,30)(23,27)(24,32)(26,28);(1,4,6,16)(2,8,10,24)(3,12,13,29)(5,19,20,15)(7,22,21,32)(9,26,17,23)(11,27,25,28)(14,31,18,30)Polycyclic presentation:
2^8|d^2=h,f^2=h,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gh,c^b=cg,e^c=eh,f^c=fh,f^d=fh40 conjugacy classes — view the full character table.
| maximal subgroup of | #139746, #144363, #144384, … |
| maximal normal subgroup of | #139746, #144363, #144384, … |
| maximal quotient of | #245065, #245068, #245073, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 16887);