Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23848 |
| automorphism | Aut(G) order 32768 |
| 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 | C2×#127 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
573 conjugacy classes of subgroups; only the 139 normal subgroups are shown.
This group has 573 conjugacy classes of subgroups — too many to draw. Only its 139 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 28 points:
(1,2)(3,4)(5,6)(7,11)(8,9)(10,12)(15,23)(16,17)(19,27)(20,21)(24,28)(25,26);(6,9)(10,11)(13,14,18,22)(15,24,26,27)(16,21,17,20)(19,23,28,25);(2,4)(13,15,18,26)(14,19,22,28)(16,25,17,23)(20,24,21,27);(5,7,8,12)(6,10,9,11)Polycyclic presentation:
2^8|b^2=h,c^2=h,d^2=e,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gh,c^b=cg,f^b=fh,g^b=gh,f^c=fh,g^c=gh52 conjugacy classes — view the full character table.
| maximal subgroup of | #244872, #244874, #244876, … |
| maximal normal subgroup of | #244872, #244874, #244876, … |
| maximal quotient of | #258873, #258886, #258895, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 17178);