Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7221 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #340 |
| commutator | C2×C8 | C42 |
| Frattini, cosocle | C23×C8 | C22 |
| prime radical | C24 | D8 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C23×C8 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
154 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)(3,7,8,5)(4,6)(10,15,16,19)(11,12)(13,21)(17,24,23,18)(20,22);(1,3)(2,5)(4,8)(6,7)(9,10,12,17,21,24,11,19,14,16,22,23,13,18,20,15)Polycyclic presentation:
2^8|a^2=d,b^2=e,c^2=fgh,e^2=fg,f^2=h,b^a=bc,c^a=cf,e^a=efg,f^a=fh,c^b=ch,d^b=dg40 conjugacy classes — view the full character table.
| maximal subgroup of | #93780, #93792, C2×#7233, … |
| maximal normal subgroup of | #93780, #93792, C2×#7233, … |
| maximal quotient of | #93021, #93025, #93028, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 221);