Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #327 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C4×C8 · Schur multiplier (H2) = C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C42 | C22⋊C4 |
| commutator | C2×C4 | C4×C8 |
| Frattini, cosocle | C2×C4×C8 | C22 |
| socle | C22 | #336 |
| prime radical | C23 | #97 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C42 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
96 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 2 generators on 48 points:
(1,2)(3,9,10,6)(4,8)(5,7)(11,15,16,13)(12,14)(17,18,21,26,22,27,36,44,23,28,37,45,20,25,34,42)(19,29,31,46,30,40,47,43,32,41,35,48,33,24,38,39);(1,3,4,11,5,10,12,16)(2,6,8,13,7,9,14,15)(17,19)(18,24)(20,33)(21,35)(22,30)(23,32)(25,41)(26,43)(27,29)(28,40)(31,37)(34,47)(36,38)(39,45)(42,46)(44,48)Polycyclic presentation:
2^8|a^2=d,b^2=e,c^2=h,d^2=g,e^2=f,g^2=h,b^a=bc,c^a=cf,c^b=ch,d^b=dfh76 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#7198, #127035, #127036, … |
| maximal normal subgroup of | C2×#7198, #127035, #127036, … |
| maximal quotient of | #93277, #93305, #93316, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 186);