Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9914 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #120 |
| commutator, Frattini, cosocle | C23×C4 | C23 |
| prime radical | C24 | C2×D4 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
316 conjugacy classes of subgroups; only the 69 normal subgroups are shown.
This group has 316 conjugacy classes of subgroups — too many to draw. Only its 69 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) = 64 |
A minimal faithful permutation representation — 3 generators on 28 points:
(1,2)(3,4)(5,6,9,7)(8,10,12,11)(13,14)(15,18)(16,19)(17,20)(21,24)(22,25)(23,27)(26,28);(6,10)(7,11)(8,12)(13,15,17,22)(14,18,20,25)(16,21,23,28)(19,26,27,24);(2,4)(5,7,9,6)(8,10,12,11)(13,16)(14,19)(15,21)(17,23)(18,24)(20,27)(22,28)(25,26)Polycyclic presentation:
2^8|a^2=g,b^2=h,c^2=g,d^2=g,f^2=g,b^a=bd,c^a=ce,d^a=dg,f^a=fh,c^b=cf,d^b=dg,f^b=fg,d^c=dh,f^c=fg40 conjugacy classes — view the full character table.
| maximal subgroup of | #93220, #125970, #125971, … |
| maximal normal subgroup of | #93220, #125970, #125971, … |
| maximal quotient of | #102439, #102454, #102467, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 2960);