Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #17367 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×C22⋊C4 |
| commutator | C23 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| prime radical | C22×#41 | C4 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C4×Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
587 conjugacy classes of subgroups; only the 199 normal subgroups are shown.
This group has 587 conjugacy classes of subgroups — too many to draw. Only its 199 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:
(1,2,3,6,4,5,7,8)(9,10,14,20,15,19,13,18)(11,17,21,24,12,16,22,23);(9,11,15,12)(10,16,19,17)(13,21,14,22)(18,23,20,24);(2,5)(6,8)(9,12,15,11)(10,17,19,16)(13,21,14,22)(18,23,20,24);(9,13,15,14)(10,18,19,20)(11,21,12,22)(16,23,17,24)Polycyclic presentation:
2^8|a^2=f,b^2=g,c^2=g,d^2=g,f^2=h,b^a=be,c^a=ch,e^a=eg,f^b=fg,d^c=dg58 conjugacy classes — view the full character table.
| maximal subgroup of | #137769, #137771, #137803, … |
| maximal normal subgroup of | #137769, #137771, #137803, … |
| maximal quotient of | #160897, #160903, #160909, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 10360);