Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23391 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C22×D4 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C43 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
373 conjugacy classes of subgroups; only the 119 normal subgroups are shown.
This group has 373 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:
(1,2,5,7)(3,8,4,6)(9,10,12,14)(11,15,16,13)(17,18,22,26)(19,23,29,32)(20,24,21,25)(27,30,28,31);(1,3,5,4)(2,6,7,8)(9,11,12,16)(10,13,14,15)(17,19)(18,23)(20,28)(21,27)(22,29)(24,31)(25,30)(26,32);(10,14)(13,15)(19,27)(20,21)(23,30)(24,25)(28,29)(31,32);(1,4,5,3)(2,6,7,8)(17,20,22,21)(18,24,26,25)(19,28,29,27)(23,31,32,30)Polycyclic presentation:
2^8|a^2=eh,b^2=e,d^2=efh,g^2=h,b^a=be,c^a=cf,c^b=cg,g^b=gh,d^c=dh,g^c=gh52 conjugacy classes — view the full character table.
| maximal subgroup of | #150349, #150353, #150687, … |
| maximal normal subgroup of | #150349, #150353, #150687, … |
| maximal quotient of | #244448, #244460, #244464, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 16814);