Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #17547 |
| automorphism | Aut(G) order 12288 |
| 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 | #3517 | C2 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C4×#41 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
508 conjugacy classes of subgroups; only the 183 normal subgroups are shown.
This group has 508 conjugacy classes of subgroups — too many to draw. Only its 183 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:
(2,5,6,7)(9,10,14,19,15,18,13,17)(11,20,21,24,12,16,22,23);(1,2)(3,7)(4,6)(5,8)(9,11,15,12)(10,16,18,20)(13,21,14,22)(17,23,19,24);(9,12,15,11)(10,16,18,20)(13,21,14,22)(17,24,19,23);(9,13,15,14)(10,17,18,19)(11,21,12,22)(16,23,20,24)Polycyclic presentation:
2^8|a^2=f,b^2=g,c^2=g,d^2=g,e^2=h,f^2=g,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 | #17655⋊C3 |
| maximal subgroup of | #221307, #221309, #221311, … |
| maximal normal subgroup of | #221307, #221309, #221311, … |
| maximal quotient of | #160507, #160511, #160514, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 10643);