Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32706 |
| 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 | C2×#127 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C43 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
407 conjugacy classes of subgroups; only the 107 normal subgroups are shown.
This group has 407 conjugacy classes of subgroups — too many to draw. Only its 107 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)(3,6)(4,5)(7,8)(9,10,12,14)(11,13,15,16)(17,18)(19,22)(20,23)(21,24)(25,32)(26,30)(27,29)(28,31);(2,5)(3,7)(9,11)(10,13)(12,15)(14,16)(17,19,20,26)(18,22,23,30)(21,25,28,27)(24,32,31,29);(3,7)(6,8)(10,14)(13,16)(18,23)(19,25)(21,28)(22,29)(26,27)(30,32);(1,3,4,7)(2,6,5,8)(11,15)(13,16)(19,26)(22,30)(25,27)(29,32)Polycyclic presentation:
2^8|a^2=fh,b^2=h,d^2=e,g^2=h,b^a=be,c^a=cf,c^b=cg,d^b=def,g^b=gh,d^c=de,g^c=gh40 conjugacy classes — view the full character table.
| maximal subgroup of | #169495, #169497, #169527, … |
| maximal normal subgroup of | #169495, #169497, #169527, … |
| maximal quotient of | #391263, #391264, #391265, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 25787);