Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2668 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C24 |
| commutator, socle | C23 | C23×C4 |
| prime radical | C22×#41 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
437 conjugacy classes of subgroups; only the 155 normal subgroups are shown.
This group has 437 conjugacy classes of subgroups — too many to draw. Only its 155 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)(3,7)(4,6)(5,8)(9,10)(11,13)(12,14)(15,16)(17,18,21,25,20,24,28,32)(19,23,27,31,26,22,29,30);(1,3)(2,5)(4,8)(6,7)(9,11,12,15)(10,13,14,16)(17,19)(18,22)(20,26)(21,27)(23,24)(25,30)(28,29)(31,32);(3,8)(5,7)(10,14)(13,16)(17,19)(18,23)(20,26)(21,27)(22,24)(25,31)(28,29)(30,32);(2,6)(5,7)(18,24)(19,26)(25,32)(27,29)Polycyclic presentation:
2^8|a^2=g,b^2=f,g^2=h,b^a=be,c^a=cf,d^a=de,c^b=ceh,d^b=dh,d^c=dh64 conjugacy classes — view the full character table.
| maximal subgroup of | #100685, #100689, #100693, … |
| maximal normal subgroup of | #100685, #100689, #100693, … |
| maximal quotient of | #355059, #355202, #355650, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 14899);