Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32884 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C24 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
365 conjugacy classes of subgroups; only the 131 normal subgroups are shown.
This group has 365 conjugacy classes of subgroups — too many to draw. Only its 131 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)(11,13)(12,14)(15,16)(17,18)(19,22)(20,23)(21,24)(25,28)(26,30)(27,29)(31,32);(1,3,5,4)(2,6,7,8)(9,11,12,15)(10,13,14,16)(17,19)(18,22)(20,26)(21,27)(23,30)(24,29)(25,31)(28,32);(10,14)(13,16)(18,23)(19,26)(24,25)(27,32);(1,4,5,3)(2,6,7,8)(18,24)(21,28)(22,29)(23,25)(27,32)(30,31)Polycyclic presentation:
2^8|a^2=e,b^2=efh,d^2=e,g^2=h,b^a=be,c^a=cf,d^a=dg,g^a=gh,c^b=ch,g^d=gh52 conjugacy classes — view the full character table.
| maximal subgroup of | #6470986, #6470989, #6470998, … |
| maximal normal subgroup of | #6470986, #6470989, #6470998, … |
| maximal quotient of | #392399, #392753, #392879, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26144);