Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #447 |
| automorphism | Aut(G) = C2×#72236479 (order 2048) · Inn(G) = C2×D4 · Out(G) = C23×#41 |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C8 | C2×D4 |
| commutator | C2×C4 | C22×C8 |
| Frattini, cosocle | C4×C8 | C23 |
| socle | C22 | C2×C4⋊C8 |
| prime radical | C23 | C4⋊C8 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
110 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 48 points:
(1,2)(3,10)(4,8)(5,14)(6,9)(7,12)(11,15)(13,16)(17,18,20,24,21,25,30,36,22,26,31,37,32,38,42,46)(19,23,28,34,29,35,41,45,27,33,39,43,40,44,47,48);(1,3,5,12,6,13,4,11)(2,7,8,10,9,15,14,16)(17,19)(18,23)(20,28)(21,29)(22,27)(24,34)(25,35)(26,33)(30,41)(31,39)(32,40)(36,45)(37,43)(38,44)(42,47)(46,48);(1,4,6,5)(2,8,9,14)(3,11,13,12)(7,10,15,16)(19,27)(23,33)(28,39)(29,40)(34,43)(35,44)(41,47)(45,48)Polycyclic presentation:
2^8|a^2=e,b^2=d,c^2=f,d^2=f,e^2=g,g^2=h,b^a=bd,d^a=df,c^b=ch76 conjugacy classes — view the full character table.
| maximal subgroup of | #135055, #135056, #135057, … |
| maximal normal subgroup of | #135055, #135056, #135057, … |
| maximal quotient of | #126670, #129087, #129094, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5002);