Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2668 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C24 |
| commutator, socle | C23 | C23×C4 |
| prime radical | C22×D4 | C2×C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
392 conjugacy classes of subgroups; only the 161 normal subgroups are shown.
This group has 392 conjugacy classes of subgroups — too many to draw. Only its 161 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) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,5)(6,8)(9,10)(11,14)(12,13)(15,16)(17,18,22,23,21,19,20,24);(1,2,4,5)(3,6,7,8)(17,19,22,24,21,18,20,23);(1,3,4,7)(2,6,5,8)(10,13)(14,16)(18,19)(23,24);(2,5)(6,8)(9,11,12,15)(10,14,13,16)(17,20,21,22)(18,23,19,24)Polycyclic presentation:
2^8|a^2=g,b^2=eg,c^2=e,d^2=f,g^2=h,b^a=be,c^a=cf,d^a=dh,c^b=ch,d^b=deh64 conjugacy classes — view the full character table.
| maximal subgroup of | #100911, #100914, #100917, … |
| maximal normal subgroup of | #100911, #100914, #100917, … |
| maximal quotient of | #356417, #356836, #357187, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 16261);