Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | Aut(G) = C2×#72286215 (order 2048) · Inn(G) = C23 · Out(G) = C26×C4 |
| homology | Abelianization (H1) = C22×C16 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C2×C16 | C23 |
| commutator, socle | C22 | C22×C16 |
| prime radical | C23 | C2×C16 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C16 ◁ G | |
| LECS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
97 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2)(3,6)(4,5)(7,8)(9,10,11,16,12,17,19,28,13,18,20,29,22,30,31,38,14,15,21,25,23,26,32,35,24,27,33,36,34,37,39,40);(2,5)(6,8);(1,3,4,7)(2,6,5,8)(10,15)(16,25)(17,26)(18,27)(28,35)(29,36)(30,37)(38,40)Polycyclic presentation:
2^8|a^2=e,c^2=d,e^2=f,f^2=g,g^2=h,b^a=bd,c^a=ch112 conjugacy classes — view the full character table.
| maximal subgroup of | #93999, #94000, #94006, … |
| maximal normal subgroup of | #93999, #94000, #94006, … |
| maximal quotient of | #151877, #151885, #151975, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6628);