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) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C22×#135 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
729 conjugacy classes of subgroups; only the 175 normal subgroups are shown.
This group has 729 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)(9,10)(11,17)(12,23)(13,18)(14,19)(15,20)(16,21)(22,24);(5,6)(7,8)(9,11)(10,16)(12,13)(14,21)(15,22)(17,18)(19,24)(20,23);(2,4)(6,8)(9,12,13,11,15,21,14,22)(10,17,19,23,20,24,18,16);(9,13,15,14)(10,18,20,19)(11,12,22,21)(16,17,23,24)Polycyclic presentation:
2^8|c^2=e,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,e^b=eh,f^b=fh,g^b=gh,d^c=dh52 conjugacy classes — view the full character table.
| maximal subgroup of | #142290, #142291, #395466, … |
| maximal normal subgroup of | #142290, #142291, #395466, … |
| maximal quotient of | #238878, #238882, #238883, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 16531);