Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #1366 |
| automorphism | Aut(G) = #72279508 (order 2048) · Inn(G) = #333 · Out(G) = #120 |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #333 |
| commutator | C42 | C2×C8 |
| Frattini, cosocle | C2×C4×C8 | C22 |
| prime radical | C23 | #100 |
| LCS | G ▷ C42 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C22×C4 ▷ C2 ▷ C1 | |
101 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)(3,7,8,5)(4,6)(9,10,12,17,13,16,25,33,14,19,26,36,28,18,37,35)(11,20,22,34,23,15,29,30,21,31,38,40,27,32,24,39);(1,3)(2,5)(4,8)(6,7)(9,11,14,21)(10,15)(12,24,26,29)(13,27,28,23)(16,20)(17,34)(18,31)(19,32)(22,25,38,37)(30,35)(33,39)(36,40)Polycyclic presentation:
2^8|a^2=d,b^2=f,c^2=fh,d^2=g,e^2=h,g^2=h,b^a=bc,c^a=ce,e^a=eh,f^a=fh,c^b=cf,d^b=def,e^b=eh,g^b=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | #94547, C2×#7135, #124129, … |
| maximal normal subgroup of | #94547, C2×#7135, #124129, … |
| maximal quotient of | #92888, #92891, #92892, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 123);