Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23924 |
| 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 | #1999 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
418 conjugacy classes of subgroups; only the 111 normal subgroups are shown.
This group has 418 conjugacy classes of subgroups — too many to draw. Only its 111 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)(4,5)(6,8)(9,10)(11,15)(12,14)(13,16)(17,18)(19,22)(20,23)(21,25)(24,28)(26,30)(27,29)(31,32);(2,5)(6,7)(9,11)(10,13)(12,16)(14,15)(17,19,21,26)(18,22,25,30)(20,27,28,32)(23,29,24,31);(11,16)(13,15)(18,23)(20,28)(22,29)(24,25)(27,32)(30,31);(1,3,4,8)(2,6,5,7)(10,14)(13,15)(19,26)(22,30)(27,32)(29,31)Polycyclic presentation:
2^8|b^2=h,d^2=eg,f^2=h,b^a=be,c^a=cf,d^a=de,f^a=fh,c^b=cg,d^b=dh,f^c=fh46 conjugacy classes — view the full character table.
| maximal subgroup of | #123017, #123027, C2×#29421, … |
| maximal normal subgroup of | #123017, #123027, C2×#29421, … |
| maximal quotient of | #363417, #371111, #371141, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 22409);