Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2218 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | C2×D8 |
| commutator | C2×C8 | C22×C4 |
| Frattini, cosocle | C22×C8 | C23 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
243 conjugacy classes of subgroups; only the 91 normal subgroups are shown.
This group has 243 conjugacy classes of subgroups — too many to draw. Only its 91 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) = 256 |
A minimal faithful permutation representation — 3 generators on 40 points:
(2,4)(5,6,7,8)(9,10)(11,19)(12,24)(13,16)(14,28)(15,22)(17,27)(18,25)(20,32)(21,34)(23,40)(26,29)(30,37)(31,36)(33,39)(35,38);(6,8)(9,11,20,36,14,23,12,22,13,21,27,39,25,38,26,37)(10,15,29,40,18,31,17,19,16,30,24,35,28,33,32,34);(1,2,3,4)(10,16)(11,21)(17,32)(18,28)(22,37)(23,38)(24,29)(36,39)Polycyclic presentation:
2^8|a^2=f,b^2=dfgh,c^2=eh,d^2=gh,g^2=h,b^a=bd,c^a=ce,d^a=dg,g^a=gh,c^b=ch52 conjugacy classes — view the full character table.
| maximal subgroup of | #131858, #131888, #136884, … |
| maximal normal subgroup of | #131858, #131888, #136884, … |
| maximal quotient of | #140524, #140573, #140902, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6195);