Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23391 |
| automorphism | Aut(G) order 131072 |
| 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 | |
571 conjugacy classes of subgroups; only the 191 normal subgroups are shown.
This group has 571 conjugacy classes of subgroups — too many to draw. Only its 191 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:
(5,6,9,11)(7,12,8,10)(13,14,16,18)(15,21,22,28)(17,25,24,20)(19,26,27,23);(1,2)(3,4)(5,7,9,8)(6,10,11,12)(14,18)(19,27)(20,25)(21,28);(2,4)(5,8,9,7)(6,10,11,12)(13,15,16,22)(14,19,18,27)(17,26,24,23)(20,28,25,21);(15,22)(19,27)(21,28)(23,26)Polycyclic presentation:
2^8|a^2=e,b^2=eh,c^2=e,f^2=h,b^a=be,c^a=cf,f^a=fh,c^b=cg,d^c=dh,f^c=fh52 conjugacy classes — view the full character table.
| maximal subgroup of | #394845, #394846, #394875, … |
| maximal normal subgroup of | #394845, #394846, #394875, … |
| maximal quotient of | #241137, #241141, #241153, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 16541);