Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11677 |
| automorphism | Aut(G) = #72305173 (order 4096) · Inn(G) = C2×#102 · Out(G) = C23×D4 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#102 |
| commutator | C2×C8 | C22×C4 |
| Frattini, cosocle | C4×C8 | C23 |
| prime radical | #466 | C4 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
161 conjugacy classes of subgroups; only the 59 normal subgroups are shown.
This group has 161 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) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2,4,7,5,3,8,6)(9,10)(11,20,23,36)(12,16)(13,27)(14,19)(15,22,31,40)(17,25)(18,26)(21,34,24,37)(28,30,38,32)(29,33)(35,39);(1,3)(2,5)(4,6)(7,8)(9,11)(10,15)(12,24)(13,28)(14,23)(16,32)(17,34)(18,30)(19,31)(20,33)(21,26)(22,39)(25,40)(27,36)(29,38)(35,37);(2,3)(6,7)(9,12,14,26)(10,16,19,18)(11,21,23,24)(13,25,29,39)(15,30,31,32)(17,33,35,27)(20,34,36,37)(22,38,40,28)Polycyclic presentation:
2^8|a^2=e,c^2=h,d^2=f,e^2=g,f^2=h,b^a=bd,c^a=cg,d^a=df,f^a=fh,c^b=cgh,d^b=dfh,e^b=eh,f^b=fh52 conjugacy classes — view the full character table.
| maximal subgroup of | #130444, #130445, #130448, … |
| maximal normal subgroup of | #130444, #130445, #130448, … |
| maximal quotient of | #125094, #125096, #125116, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4670);