Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23391 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C24 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
551 conjugacy classes of subgroups; only the 179 normal subgroups are shown.
This group has 551 conjugacy classes of subgroups — too many to draw. Only its 179 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:
(1,2,3,4)(9,10,15,17)(11,14,19,13)(12,20,23,18)(16,21,24,22);(5,6)(7,8)(9,11,14,17,15,19,13,10)(12,16,22,20,23,24,21,18);(2,4)(6,8)(9,12,15,23)(10,16,17,24)(11,18,19,20)(13,22,14,21);(9,13,15,14)(10,11,17,19)(12,21,23,22)(16,20,24,18)Polycyclic presentation:
2^8|a^2=ef,b^2=e,c^2=h,d^2=h,e^2=h,f^2=h,g^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fh,g^a=gh,c^b=cg,d^b=dh,d^c=dh,e^c=eh,f^c=fh,g^c=gh52 conjugacy classes — view the full character table.
| maximal subgroup of | #395728, #395903, #395905, … |
| maximal normal subgroup of | #395728, #395903, #395905, … |
| maximal quotient of | #242461, #242464, #242466, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 16607);