Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | Aut(G) = #72218147 (order 8192) · Inn(G) = C22 · Out(G) = #72218016 |
| homology | Abelianization (H1) = C8×C16 · Schur multiplier (H2) = C8 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C82 | C22 |
| commutator | C2 | C8×C16 |
| socle, prime radical | C22 | C82 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C82 ◁ G | |
| LECS | G ▷ C82 ▷ C42 ▷ C22 ▷ C1 | |
75 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 2 generators on 32 points:
(1,2,3,7,4,8,9,14,5,6,10,12,11,13,15,16);(2,6)(7,12)(8,13)(14,16)(17,18,19,22,20,23,25,28,21,24,26,29,27,30,31,32)Polycyclic presentation:
2^8|a^2=c,b^2=d,c^2=e,d^2=f,e^2=g,f^2=h,b^a=bg160 conjugacy classes — view the full character table.
| Frattini subgroup of | #72230695, #72230713, #72230715 |
| maximal subgroup of | #93666, #93667, #93752, … |
| maximal normal subgroup of | #93666, #93667, #93752, … |
| maximal quotient of | #92895, #93665, #93666, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 40);