Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #17547 |
| automorphism | Aut(G) order 24576 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×C22⋊C4 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C42 | C24 |
| prime radical | C2×#41 | D4 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C4×#41 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
482 conjugacy classes of subgroups; only the 225 normal subgroups are shown.
This group has 482 conjugacy classes of subgroups — too many to draw. Only its 225 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:
(1,2,5,8,6,3,4,7)(9,10,12,16,13,14,19,21)(11,15,18,22,17,20,23,24);(1,3,6,2)(4,7,5,8)(10,14)(12,19)(15,20)(18,23);(9,11)(10,15)(12,18)(13,17)(14,20)(16,22)(19,23)(21,24);(11,17)(15,20)(18,23)(22,24)Polycyclic presentation:
2^8|a^2=f,b^2=h,e^2=h,f^2=gh,b^a=be,e^a=eg,e^b=eh,f^b=fgh,d^c=dg61 conjugacy classes — view the full character table.
| Fitting subgroup of | #17722⋊C3 |
| maximal subgroup of | #221232, #221233, #221236, … |
| maximal normal subgroup of | #221232, #221233, #221236, … |
| maximal quotient of | #160508, #160511, #161094, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 10710);