Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #20671 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | C2×#117 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| socle | C2 | C2×#380 |
| prime radical | C22×#41 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×M16 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
662 conjugacy classes of subgroups; only the 261 normal subgroups are shown.
This group has 662 conjugacy classes of subgroups — too many to draw. Only its 261 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) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,11)(4,15,17,8)(5,18)(6,10)(7,14)(9,19)(12,25,26,23)(13,24)(16,22,31,30)(20,27,29,32)(21,28);(1,3,5,14,6,13,19,28)(2,7,9,11,10,21,18,24)(4,12,16,27,17,26,31,32)(8,20,22,23,15,29,30,25);(1,4)(2,8)(3,12)(5,16)(6,17)(7,20)(9,22)(10,15)(11,23)(13,26)(14,27)(18,30)(19,31)(21,29)(24,25)(28,32);(3,13)(7,21)(11,24)(12,26)(14,28)(20,29)(23,25)(27,32)Polycyclic presentation:
2^8|a^2=g,b^2=e,e^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fh,d^b=dh,g^c=gh58 conjugacy classes — view the full character table.
| maximal subgroup of | #138354, #138355, #148017, … |
| maximal normal subgroup of | #138354, #138355, #148017, … |
| maximal quotient of | #188755, #188756, #188759, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 13659);