Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3186 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22×D4 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C42 | C24 |
| socle | C22 | C22×C4⋊C4 |
| prime radical | C24 | C4⋊C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
365 conjugacy classes of subgroups; only the 227 normal subgroups are shown.
This group has 365 conjugacy classes of subgroups — too many to draw. Only its 227 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:
(2,5)(3,6)(7,8)(9,10,12,16,13,15,19,22)(11,14,18,21,17,20,23,24);(1,2,6,8,4,7,3,5)(9,11)(10,14)(12,18)(13,17)(15,20)(16,21)(19,23)(22,24);(11,17)(14,20)(18,23)(21,24);(10,15)(14,20)(16,22)(21,24)Polycyclic presentation:
2^8|a^2=f,b^2=eg,e^2=g,f^2=h,b^a=be,d^a=dh,e^a=eg,c^b=ch64 conjugacy classes — view the full character table.
| maximal subgroup of | #109720, #109724, #109788, … |
| maximal normal subgroup of | #109720, #109724, #109788, … |
| maximal quotient of | #180926, #183392, #183408, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 10802);