Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3057 |
| automorphism | Aut(G) order 262144 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C42 |
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 | C23×D8 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
895 conjugacy classes of subgroups; only the 247 normal subgroups are shown.
This group has 895 conjugacy classes of subgroups — too many to draw. Only its 247 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,4)(5,6,7,8)(9,10)(11,13)(12,18)(14,19)(15,17)(16,21)(20,23)(22,24);(1,2,3,4)(6,8)(9,11,13,10,15,19,14,17)(12,18,20,16,22,24,21,23);(9,12,13,20,15,22,14,21)(10,16,19,24,17,23,11,18);(9,13,15,14)(10,11,17,19)(12,20,22,21)(16,18,23,24)Polycyclic presentation:
2^8|a^2=g,b^2=egh,c^2=fh,d^2=h,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 | #124263, #124940, #140617, … |
| maximal normal subgroup of | #124263, #124940, #140617, … |
| maximal quotient of | #204266, #204283, #204300, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 14221);