Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #1414 |
| automorphism | Aut(G) = #72284457 (order 2048) · Inn(G) = #354 · Out(G) = #120 |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #354 |
| commutator | C42 | C2×C8 |
| Frattini, cosocle | #382 | C22 |
| prime radical | C23 | #99 |
| LCS | G ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23 ◁ C22×C4 ◁ #382 ◁ G | |
| LECS | G ▷ #382 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
88 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 2 generators on 40 points:
(1,2,3,5,4,6,7,8)(9,10)(11,19,21,34)(12,23,25,39)(13,27,29,17)(14,18)(15,20,30,37)(16,26)(22,36,28,35)(24,31,38,32)(33,40);(9,11)(10,15,18,32)(12,24)(13,28)(14,22)(16,19,33,36)(17,30,27,31)(20,25)(21,29)(23,34,39,35)(26,38)(37,40)Polycyclic presentation:
2^8|a^2=d,b^2=g,c^2=g,d^2=f,e^2=h,b^a=bc,c^a=ce,e^a=eg,g^a=gh,c^b=cgh,d^b=degh,e^b=eh,d^c=dgh,e^d=eh34 conjugacy classes — view the full character table.
| maximal subgroup of | #94547, C2×#7359, #149586, … |
| maximal normal subgroup of | #94547, C2×#7359, #149586, … |
| maximal quotient of | #93698, #93716, #93911, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 347);