Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3008 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
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 | C24 | C2×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
691 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)(10,14)(11,17)(12,19)(16,20)(18,21)(22,24);(2,4)(5,7,8,6)(9,10)(11,15)(12,20)(13,16)(14,19)(17,22)(18,23)(21,24);(9,11,13,18)(10,15,16,23)(12,21,19,17)(14,22,20,24);(10,16)(14,20)(15,23)(22,24)Polycyclic presentation:
2^8|a^2=g,b^2=g,c^2=h,e^2=h,f^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fh,d^b=dh,e^b=eh,f^b=fh,e^c=eh,f^c=fh64 conjugacy classes — view the full character table.
| maximal subgroup of | #137731, #137733, #137799, … |
| maximal normal subgroup of | #137731, #137733, #137799, … |
| maximal quotient of | #191867, #191882, #191897, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 13865);