Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3055 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| prime radical | C24 | C2×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
561 conjugacy classes of subgroups; only the 211 normal subgroups are shown.
This group has 561 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) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,4)(5,6,8,7)(9,10,15,20)(11,17,22,23)(12,16,21,24)(13,18,14,19);(5,7,8,6)(9,11,14,21,15,22,13,12)(10,16,18,23,20,24,19,17);(2,4)(9,12,13,22,15,21,14,11)(10,17,19,24,20,23,18,16);(9,13,15,14)(10,18,20,19)(11,21,22,12)(16,17,24,23)Polycyclic presentation:
2^8|a^2=g,b^2=eg,c^2=e,d^2=h,e^2=h,f^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fh,d^b=dh,d^c=dh64 conjugacy classes — view the full character table.
| maximal subgroup of | #368469, #368472, #368508, … |
| maximal normal subgroup of | #368469, #368472, #368508, … |
| maximal quotient of | #202549, #202564, #202579, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 14082);