Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #17392 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
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 | C25 | C2×C4 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
693 conjugacy classes of subgroups; only the 195 normal subgroups are shown.
This group has 693 conjugacy classes of subgroups — too many to draw. Only its 195 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) = 32 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,7,8,5)(4,6)(9,10,13,18,14,17,12,16)(11,15,19,22,20,23,21,24);(1,3,4,8)(2,5,6,7)(9,11)(10,15)(12,21)(13,19)(14,20)(16,24)(17,23)(18,22);(9,12,14,13)(10,16,17,18)(11,19,20,21)(15,22,23,24);(10,17)(15,23)(16,18)(22,24)Polycyclic presentation:
2^8|a^2=f,b^2=g,c^2=h,f^2=h,b^a=be,d^a=dh,e^a=eg,c^b=ch,f^b=fg52 conjugacy classes — view the full character table.
| maximal subgroup of | #105754, #106090, #107205, … |
| maximal normal subgroup of | #105754, #106090, #107205, … |
| maximal quotient of | #162579, #162584, #162589, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 10385);