Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #20671 |
| automorphism | Aut(G) order 16384 |
| 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×#381 |
| prime radical | C22×#41 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
630 conjugacy classes of subgroups; only the 221 normal subgroups are shown.
This group has 630 conjugacy classes of subgroups — too many to draw. Only its 221 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)(3,11,13,7)(4,15,17,30)(5,10)(6,9)(8,16,21,31)(12,25)(14,24,28,20)(18,22)(19,27)(23,32)(26,29);(1,3)(2,7)(4,12)(5,14)(6,13)(8,19)(9,11)(10,20)(15,29)(16,27)(17,26)(18,28)(21,23)(22,24)(25,30)(31,32);(1,4)(2,8,9,21)(3,12)(5,16)(6,17)(7,19,11,23)(10,15,22,30)(13,26)(14,27)(18,31)(20,29,24,25)(28,32);(3,13)(7,11)(12,26)(14,28)(19,23)(20,24)(25,29)(27,32)Polycyclic presentation:
2^8|a^2=g,c^2=e,f^2=h,b^a=be,c^a=cf,e^a=eh,d^b=dh,g^b=gh,g^c=gh58 conjugacy classes — view the full character table.
| maximal subgroup of | #147612, #147613, #147614, … |
| maximal normal subgroup of | #147612, #147613, #147614, … |
| maximal quotient of | #189989, #189990, #189994, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 13696);