Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23387 |
| 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 | C22×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
551 conjugacy classes of subgroups; only the 175 normal subgroups are shown.
This group has 551 conjugacy classes of subgroups — too many to draw. Only its 175 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)(10,15)(11,19)(12,13)(16,23)(17,18)(21,22);(5,6)(7,8)(9,10,13,17,14,18,12,15)(11,20,19,23,21,24,22,16);(2,4)(6,8)(9,11)(10,16)(12,19)(13,22)(14,21)(15,20)(17,24)(18,23);(11,21)(16,23)(19,22)(20,24)Polycyclic presentation:
2^8|a^2=ef,b^2=eh,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^c=dh,e^c=eh,f^c=fh,g^c=gh52 conjugacy classes — view the full character table.
| maximal subgroup of | #395663, #395731, #395732, … |
| maximal normal subgroup of | #395663, #395731, #395732, … |
| maximal quotient of | #242420, #242424, #242428, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 16599);