Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1304 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C4×C8 · Schur multiplier (H2) = C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #95 |
| commutator, prime radical | C23 | C4×C8 |
| Frattini, cosocle | C2×C4×C8 | C22 |
| socle | C22 | #336 |
| LCS | G ▷ C23 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
92 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)(17,18,20,24,21,26,31,38,22,25,32,37,33,39,44,47)(19,27,29,40,30,41,43,48,28,23,34,35,42,36,45,46);(1,3,4,11,5,10,12,16)(2,6,8,13,7,9,14,15)(17,19)(18,23,25,27)(20,29)(21,30)(22,28)(24,35,37,40)(26,36,39,41)(31,43)(32,34)(33,42)(38,46,47,48)(44,45)Polycyclic presentation:
2^8|a^2=d,b^2=e,d^2=g,e^2=f,g^2=h,b^a=bc,c^a=cf,e^a=eh,c^b=ch,d^b=df52 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#7197, #127032, #127033, … |
| maximal normal subgroup of | C2×#7197, #127032, #127033, … |
| maximal quotient of | #93276, #93305, #93316, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 185);