Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3264 |
| automorphism | Aut(G) = C211×D4 (order 16384) · Inn(G) = C22×D4 · Out(G) = C29 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| prime radical | C2×#121 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
499 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,4)(5,6,8,7)(9,10,12,17,13,15,21,14)(11,18,19,22,20,16,23,24);(5,7,8,6)(10,14)(11,19)(12,21)(15,17)(20,23)(22,24);(2,4)(10,15)(11,20)(14,17)(19,23);(9,11,21,23,13,20,12,19)(10,16,14,22,15,18,17,24)Polycyclic presentation:
2^8|a^2=g,b^2=egh,d^2=eh,e^2=h,g^2=h,b^a=be,c^a=cf,d^a=dh,d^b=de,e^b=eh,g^b=gh,d^c=dh58 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#23041, #6381944, #6381953, … |
| maximal normal subgroup of | C2×#23041, #6381944, #6381953, … |
| maximal quotient of | #229121, #229140, #229159, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 16029);