Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7217 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #340 |
| commutator | C2×C8 | C42 |
| Frattini, cosocle | C2×C4×C8 | C22 |
| prime radical | C23 | #107 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
95 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 2 generators on 40 points:
(1,2,3,6,4,5,7,8)(9,10)(11,21,23,35)(12,18)(13,28)(14,30)(15,20)(16,22,31,36)(17,26)(19,29)(24,39,37,33)(25,40,38,32)(27,34);(2,5)(3,7)(9,11,13,24,29,38,27,36,15,23,26,37,14,25,12,22)(10,16,18,32,30,39,17,21,20,31,34,40,19,33,28,35)Polycyclic presentation:
2^8|a^2=d,b^2=e,c^2=fgh,d^2=gh,e^2=fg,f^2=h,b^a=bc,c^a=cf,e^a=efgh,f^a=fh,d^b=dg52 conjugacy classes — view the full character table.
| maximal subgroup of | #93848, #93849, #93856, … |
| maximal normal subgroup of | #93848, #93849, #93856, … |
| maximal quotient of | #93329, #93347, #93636, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 230);