Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | Aut(G) = C2×#72285493 (order 2048) · Inn(G) = D4 · Out(G) = C2×#2397 |
| homology | Abelianization (H1) = C4×C16 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C16 | D4 |
| commutator | C4 | C4×C16 |
| Frattini, cosocle | C22×C16 | C22 |
| socle | C22 | C4⋊C16 |
| prime radical | C23 | M32 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C16 ◁ C22×C16 ◁ G | |
| LECS | G ▷ C22×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
95 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 2 generators on 36 points:
(1,2)(3,4)(5,6,8,14,9,16,19,29,10,13,20,26,22,27,32,35,11,15,21,28,23,30,33,36,7,12,17,24,18,25,31,34);(2,4)(6,12,15,13)(14,24,28,26)(16,25,30,27)(29,34,36,35)Polycyclic presentation:
2^8|a^2=d,b^2=e,c^2=h,d^2=f,f^2=g,g^2=h,b^a=bc,c^a=ch,e^a=eh112 conjugacy classes — view the full character table.
| maximal subgroup of | #93889, #93890, #93918, … |
| maximal normal subgroup of | #93889, #93890, #93918, … |
| maximal quotient of | #93991, #93999, #94006, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 455);