Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11858 |
| automorphism | Aut(G) = #72306287 (order 4096) · Inn(G) = C2×#102 · Out(G) = C22×#41 |
| 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 | #3050 | C2 |
| LCS | G ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C23×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
197 conjugacy classes of subgroups; only the 57 normal subgroups are shown.
This group has 197 conjugacy classes of subgroups — too many to draw. Only its 57 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)(9,10,14,20,16,22,15,21)(11,23,26,38,28,39,27,34)(12,29,13,32,31,19,30,18)(17,33,35,25,37,24,36,40);(1,3)(2,5)(4,8)(6,7)(9,11)(10,17)(12,25)(13,33)(14,27)(15,26)(16,28)(18,23)(19,38)(20,36)(21,35)(22,37)(24,30)(29,34)(31,40)(32,39);(3,8)(5,7)(9,12,15,30,16,31,14,13)(10,18,20,29,22,32,21,19)(11,24,27,25,28,33,26,40)(17,34,35,23,37,38,36,39)Polycyclic presentation:
2^8|a^2=e,c^2=g,d^2=gh,e^2=h,f^2=h,g^2=h,b^a=bd,c^a=cg,d^a=df,f^a=fh,g^a=gh,c^b=cfh,d^b=dg,e^b=efgh,f^b=fh,g^b=gh40 conjugacy classes — view the full character table.
| maximal subgroup of | #130242, #130243, #130254, … |
| maximal normal subgroup of | #130242, #130243, #130254, … |
| maximal quotient of | #122744, #122745, #122750, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4882);