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 | C22×#41 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
729 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)(9,10,14,18)(11,19,22,23)(12,15,13,17)(16,21,24,20);(2,4)(5,7,8,6)(10,15)(12,13)(16,23)(17,18)(19,24)(20,21);(9,11)(10,16)(12,20)(13,21)(14,22)(15,23)(17,19)(18,24);(9,12,14,13)(10,17,18,15)(11,20,22,21)(16,19,24,23)Polycyclic presentation:
2^8|a^2=g,b^2=gh,d^2=h,e^2=h,f^2=h,b^a=be,c^a=cf,d^a=dh,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 | #137734, #137737, #137802, … |
| maximal normal subgroup of | #137734, #137737, #137802, … |
| maximal quotient of | #191872, #191887, #191902, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 13870);