Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9914 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
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 | |
250 conjugacy classes of subgroups; only the 61 normal subgroups are shown.
This group has 250 conjugacy classes of subgroups — too many to draw. Only its 61 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,4,6)(3,7,8,5)(9,10,11,13)(12,14,16,15)(17,18)(19,22)(20,23)(21,24)(25,30)(26,29)(27,31)(28,32);(1,3,4,8)(2,5,6,7)(10,13)(14,15)(17,19)(18,22)(20,25)(21,26)(23,30)(24,29)(27,32)(28,31);(10,14)(12,16)(13,15)(17,20,21,27)(18,23,24,31)(19,25,26,32)(22,28,29,30)Polycyclic presentation:
2^8|a^2=d,b^2=dg,c^2=h,e^2=g,b^a=bd,c^a=ce,e^a=eg,f^a=fh,c^b=cf,e^b=eh,e^c=eg40 conjugacy classes — view the full character table.
| maximal subgroup of | #93239, C2×#10346, #237736, … |
| maximal normal subgroup of | #93239, C2×#10346, #237736, … |
| maximal quotient of | #104716, #104721, #104723, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 3334);