Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3057 |
| automorphism | Aut(G) order 131072 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
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×#135 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
783 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,4)(5,6,7,8)(10,14)(11,17)(12,18)(15,22)(16,21)(19,23);(1,2,3,4)(6,8)(9,10,12,14,13,16,18,21)(11,15,17,20,19,22,23,24);(9,11,18,23,13,19,12,17)(10,15,21,24,16,22,14,20);(10,16)(14,21)(15,22)(20,24)Polycyclic presentation:
2^8|a^2=g,b^2=eg,c^2=fh,e^2=h,f^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fh,d^b=dh64 conjugacy classes — view the full character table.
| maximal subgroup of | #147422, #147920, #147921, … |
| maximal normal subgroup of | #147422, #147920, #147921, … |
| maximal quotient of | #204263, #204280, #204297, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 14218);