Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1284 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #95 |
| commutator | C2×C4 | C2×C42 |
| Frattini, cosocle | C2×C42 | C23 |
| socle | C22 | C2×#95 |
| prime radical | C24 | C22⋊C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
328 conjugacy classes of subgroups; only the 123 normal subgroups are shown.
This group has 328 conjugacy classes of subgroups — too many to draw. Only its 123 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 — 3 generators on 24 points:
(1,2,4,7,5,8,3,6)(9,10)(11,17,19,23)(12,15)(13,16)(14,18,21,24)(20,22);(2,6,8,7)(9,11)(10,14,16,21)(12,18)(13,19)(15,17,22,23)(20,24);(1,3,5,4)(2,6,8,7)Polycyclic presentation:
2^8|a^2=e,b^2=f,c^2=g,d^2=gh,e^2=g,b^a=bd,d^a=dg,f^a=fgh,e^b=eh64 conjugacy classes — view the full character table.
| maximal subgroup of | #105398, #105399, #105654, … |
| maximal normal subgroup of | #105398, #105399, #105654, … |
| maximal quotient of | #94955, #94958, #94968, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 936);