Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11704 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#102 |
| commutator | C2×C8 | C22×C4 |
| Frattini, cosocle | C22×C8 | C23 |
| prime radical | #3169 | C2 |
| LCS | G ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
182 conjugacy classes of subgroups; only the 59 normal subgroups are shown.
This group has 182 conjugacy classes of subgroups — too many to draw. Only its 59 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 24 points:
(2,6,7,8)(9,10,11,16,14,19,13,18)(12,21,20,17,23,15,22,24);(1,2)(3,6)(4,8)(5,7)(10,15)(11,13)(12,22)(16,17)(18,24)(19,21)(20,23);(1,3,5,4)(2,6,7,8)(9,11,14,13)(10,16,19,18)(12,20,23,22)(15,24,21,17)Polycyclic presentation:
2^8|a^2=e,c^2=fgh,d^2=gh,e^2=h,f^2=h,g^2=h,b^a=bd,d^a=df,f^a=fh,g^a=gh,c^b=ch,d^b=dg,e^b=efgh,f^b=fh,g^b=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | #129015, #129033, #131664, … |
| maximal normal subgroup of | #129015, #129033, #131664, … |
| maximal quotient of | #124110, #124220, #124235, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4739);