Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30412 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C22×D8 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
413 conjugacy classes of subgroups; only the 115 normal subgroups are shown.
This group has 413 conjugacy classes of subgroups — too many to draw. Only its 115 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:
(5,6,9,12)(7,8,11,10)(13,14,17,21,16,18,25,19)(15,22,24,27,23,28,26,20);(1,2)(3,4)(5,7,9,11)(6,10,12,8)(14,18)(15,23)(19,21)(24,26);(2,4)(5,8,9,10)(6,11,12,7)(14,19)(17,25)(18,21)(20,22)(24,26)(27,28);(6,12)(7,11)(13,15,16,23)(14,20,18,27)(17,26,25,24)(19,22,21,28)Polycyclic presentation:
2^8|a^2=efh,b^2=eh,c^2=eh,d^2=h,f^2=h,b^a=be,c^a=cf,d^a=defh,c^b=cg,d^b=de,f^c=fh,f^d=fh40 conjugacy classes — view the full character table.
| maximal subgroup of | #125710, #126047, #126391, … |
| maximal normal subgroup of | #125710, #126047, #126391, … |
| maximal quotient of | #345149, #345157, #345163, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 24215);