Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1462 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×C22⋊C4 |
| commutator | C23 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| socle | C22 | C22×C22⋊C4 |
| prime radical | C24 | C22×C4 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
607 conjugacy classes of subgroups; only the 211 normal subgroups are shown.
This group has 607 conjugacy classes of subgroups — too many to draw. Only its 211 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,4,8,5,6,3,7)(9,10)(11,17,19,14)(12,15)(13,16)(18,23,24,21)(20,22);(9,11,13,19)(10,14,16,17)(12,18,20,24)(15,21,22,23);(2,6)(7,8);(1,3,5,4)(2,7,6,8)(9,12,13,20)(10,15,16,22)(11,18,19,24)(14,21,17,23)Polycyclic presentation:
2^8|a^2=f,b^2=g,d^2=gh,f^2=h,b^a=be,c^a=ch,e^a=eg,f^b=fg64 conjugacy classes — view the full character table.
| maximal subgroup of | #105583, #105593, #105619, … |
| maximal normal subgroup of | #105583, #105593, #105619, … |
| maximal quotient of | #160712, #160718, #160723, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 10353);