Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7213 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #342 |
| commutator | C22×C4 | C42 |
| Frattini, cosocle | C2×C4×D4 | C22 |
| prime radical | C22×D4 | Q8 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C2×C4×D4 ▷ C23 ▷ C2 ▷ C1 | |
156 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 2 generators on 24 points:
(1,2)(3,6,8,5)(4,7)(9,10,12,17,14,18,13,16)(11,19,20,15,22,23,21,24);(2,5,7,6)(9,11,13,21,14,22,12,20)(10,15,17,23,18,24,16,19)Polycyclic presentation:
2^8|a^2=d,b^2=e,c^2=fg,d^2=fgh,e^2=fgh,b^a=bc,c^a=cf,e^a=efg,f^a=fh,g^a=gh,d^b=dg,d^c=dh40 conjugacy classes — view the full character table.
| maximal subgroup of | #93828, #93829, #93831, … |
| maximal normal subgroup of | #93828, #93829, #93831, … |
| maximal quotient of | #93336, #93341, #93629, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 237);