Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10120 |
| automorphism | Aut(G) = C2×#72278177 (order 4096) · Inn(G) = #392 · Out(G) = C2×#120 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #392 |
| commutator, Frattini, cosocle | C2×C42 | C23 |
| prime radical | #3153 | C2 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
272 conjugacy classes of subgroups; only the 49 normal subgroups are shown.
This group has 272 conjugacy classes of subgroups — too many to draw. Only its 49 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:
(2,5)(3,7)(6,8)(9,10)(11,20)(12,25)(13,30)(14,18)(15,19)(16,22)(17,28)(21,38)(23,33)(24,32)(26,37)(27,36)(29,35)(31,39)(34,40);(3,7)(5,8)(9,11)(10,16)(12,26)(13,23)(14,22)(15,24)(17,34)(18,32)(19,33)(20,30)(21,27)(25,31)(28,39)(29,40)(35,38)(36,37);(1,2)(3,8)(4,6)(5,7)(9,12,15,29)(10,17,19,36)(11,21,24,39)(13,27,14,28)(16,31,33,38)(18,35,30,25)(20,37,32,34)(22,26,23,40)Polycyclic presentation:
2^8|c^2=g,d^2=g,e^2=gh,f^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=egh,c^b=cf,d^b=dg,e^b=eh,f^b=fg,d^c=dh,e^c=egh,f^c=fg31 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#10121, #236601, #236611, … |
| maximal normal subgroup of | C2×#10121, #236601, #236611, … |
| maximal quotient of | #102507, #102508, #102515, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 3109);