Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2218 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×D8 |
| commutator | C2×C8 | C22×C4 |
| Frattini, cosocle | C22×C8 | C23 |
| socle | C22 | C4×D8 |
| prime radical | C23 | #135 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C8 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
139 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(1,2)(3,4)(5,6,10,15,11,17,26,16)(7,12,9,25,19,14,27,29)(8,20,23,34,24,28,33,35)(13,21,30,36,32,18,31,22);(2,4)(5,7,10,9,11,19,26,27)(6,12,15,25,17,14,16,29)(8,21,23,36,24,18,33,22)(13,20,30,34,32,28,31,35);(5,8,19,21,10,23,27,36,11,24,7,18,26,33,9,22)(6,13,25,35,16,31,12,28,17,32,29,34,15,30,14,20)Polycyclic presentation:
2^8|a^2=f,b^2=f,c^2=eg,e^2=gh,f^2=h,g^2=h,b^a=bd,c^a=ce,e^a=eg,g^a=gh,c^b=ch52 conjugacy classes — view the full character table.
| maximal subgroup of | #93936, #93937, #152442, … |
| maximal normal subgroup of | #93936, #93937, #152442, … |
| maximal quotient of | #140343, #140344, #140363, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6158);