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) = C23 |
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 | C24 | SD16 |
| LCS | G ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C23×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
220 conjugacy classes of subgroups; only the 65 normal subgroups are shown.
This group has 220 conjugacy classes of subgroups — too many to draw. Only its 65 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2)(3,7,8,5)(4,6)(9,10,12,17,15,20,14,19)(11,21,22,35,25,38,24,37)(13,27,26,18,30,34,29,36)(16,28,31,23,33,39,32,40);(1,3)(2,5)(4,8)(6,7)(9,11,15,25)(10,16,20,33)(12,24,14,22)(13,28,30,39)(17,32,19,31)(18,35,36,37)(21,27,38,34)(23,26,40,29);(9,12,15,14)(10,17,20,19)(11,22,25,24)(13,26,30,29)(16,31,33,32)(18,34,36,27)(21,35,38,37)(23,39,40,28)Polycyclic presentation:
2^8|a^2=e,b^2=h,c^2=h,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 | #128809, #128837, #129773, … |
| maximal normal subgroup of | #128809, #128837, #129773, … |
| maximal quotient of | #122586, #122588, #122591, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4700);