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 | C22×D4 | C23 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
491 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,4,3)(5,6,8,7)(9,10,12,16)(11,18,19,14)(13,15,21,17)(20,23,24,22);(1,3,4,2)(6,7)(9,11,12,19)(10,14,16,18)(13,20,21,24)(15,22,17,23);(10,15)(13,21)(14,22)(16,17)(18,23)(20,24);(5,7,8,6)(10,16)(11,19)(15,17)(20,24)Polycyclic presentation:
2^8|a^2=g,b^2=egh,d^2=eh,f^2=h,b^a=be,c^a=cf,d^a=dh,f^a=fh,d^b=de,f^c=fh58 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#23012, #6381850, #6381853, … |
| maximal normal subgroup of | C2×#23012, #6381850, #6381853, … |
| maximal quotient of | #229627, #229635, #229659, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 16000);