Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23918 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C43 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
326 conjugacy classes of subgroups; only the 137 normal subgroups are shown.
This group has 326 conjugacy classes of subgroups — too many to draw. Only its 137 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) = 256 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,5,8)(3,7,4,6)(17,18,20,24)(19,26,27,22)(21,23,29,25)(28,31,32,30);(1,3,5,4)(2,6,8,7)(9,10,12,14)(11,15,16,13)(17,19,20,27)(18,22,24,26)(21,28,29,32)(23,30,25,31);(9,11,12,16)(10,13,14,15)(18,23)(21,29)(22,30)(24,25)(26,31)(28,32);(1,4,5,3)(2,7,8,6)(18,24)(22,26)(23,25)(30,31)Polycyclic presentation:
2^8|a^2=e,b^2=eg,c^2=g,d^2=eh,f^2=h,b^a=be,c^a=cf,d^a=de,f^a=fh,c^b=cg,f^c=fh52 conjugacy classes — view the full character table.
| maximal subgroup of | #123136, #123146, C2×#29291, … |
| maximal normal subgroup of | #123136, #123146, C2×#29291, … |
| maximal quotient of | #362941, #368278, #368279, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 22279);