Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #365 |
| automorphism | Aut(G) = #72283942 (order 1024) · Inn(G) = #106 · Out(G) = C2×#41 |
| homology | Abelianization (H1) = C2×C16 · Schur multiplier (H2) = C1 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C8 | #106 |
| commutator | C8 | C2×C16 |
| Frattini, cosocle | C4×C16 | C22 |
| socle | C2 | #1367 |
| prime radical | C22 | #364 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8 ◁ C2×C16 ◁ C4×C16 ◁ G | |
| LECS | G ▷ C4×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
44 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 2 generators on 32 points:
(1,2,5,12,7,11,18,28,8,13,16,27,4,20,19,31,9,14,24,30,21,29,17,32,6,25,23,10,22,26,3,15);(1,3,4,16,6,17,7,5,9,19,22,23,8,18,21,24)(2,10,11,15,13,12,20,28,14,27,29,31,25,30,26,32)Polycyclic presentation:
2^8|a^2=d,b^2=c,c^2=e,d^2=f,e^2=h,f^2=g,g^2=h,b^a=bc,c^a=ce,e^a=eh,d^b=dh64 conjugacy classes — view the full character table.
| maximal subgroup of | #93759, #93860, #93861, … |
| maximal normal subgroup of | #93759, #93860, #93861, … |
| maximal quotient of | #93876, #94019, C2×#7456, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 444);