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, prime radical | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #376 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
323 conjugacy classes of subgroups; only the 135 normal subgroups are shown.
This group has 323 conjugacy classes of subgroups — too many to draw. Only its 135 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)(19,22)(20,23)(21,25)(24,28)(26,30)(27,29)(31,32);(1,3,5,4)(2,6,8,7)(9,10,12,14)(11,15,16,13)(17,19,21,26)(18,22,25,30)(20,27,28,32)(23,29,24,31);(9,11,12,16)(10,13,14,15)(18,23)(20,28)(22,29)(24,25)(27,32)(30,31);(1,4,5,3)(2,7,8,6)(19,26)(22,30)(27,32)(29,31)Polycyclic presentation:
2^8|a^2=e,b^2=egh,c^2=g,d^2=e,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 | #123176, #123186, C2×#29293, … |
| maximal normal subgroup of | #123176, #123186, C2×#29293, … |
| maximal quotient of | #362946, #368282, #368283, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 22281);