Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30412 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C2×C42 |
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 ◁ C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
358 conjugacy classes of subgroups; only the 111 normal subgroups are shown.
This group has 358 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:
(2,5)(3,7)(9,10)(11,15)(12,14)(13,16)(17,18,21,23)(19,22,27,30)(20,24,28,32)(25,29,26,31);(1,2,4,5)(3,8,7,6)(17,19)(18,22)(20,25)(21,27)(23,30)(24,29)(26,28)(31,32);(9,11,12,16)(10,13,14,15)(19,25)(20,28)(22,29)(24,32)(26,27)(30,31);(1,3,4,7)(2,6,5,8)(10,14)(13,15)(18,23)(22,30)(24,32)(29,31)Polycyclic presentation:
2^8|a^2=h,b^2=e,c^2=f,d^2=e,g^2=h,b^a=be,c^a=cf,d^a=defh,c^b=cg,d^b=de,g^b=gh,g^c=gh40 conjugacy classes — view the full character table.
| maximal subgroup of | #123254, #123270, #6637153, … |
| maximal normal subgroup of | #123254, #123270, #6637153, … |
| maximal quotient of | #384381, #385691, #386215, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 25112);