Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #13694 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×D16 |
| commutator, Frattini, cosocle | C2×C16 | C23 |
| prime radical | #2246 | C2 |
| LCS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ C4×C16 ◁ G | |
| LECS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
125 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 36 points:
(2,4)(6,11)(7,16)(8,17)(9,20)(12,19)(13,28)(14,31)(15,25)(18,27)(21,34)(22,23)(24,29)(26,35)(30,36)(32,33);(5,6,12,26,8,13,18,24,20,29,33,28,34,35,7,11,10,15,16,23,21,30,32,31,9,14,27,36,17,22,19,25);(1,2,3,4)Polycyclic presentation:
2^8|b^2=dfh,c^2=e,d^2=fg,f^2=gh,g^2=h,b^a=bd,c^a=ce,d^a=df,f^a=fg,g^a=gh70 conjugacy classes — view the full character table.
| maximal subgroup of | #151514, #151515, #151516, … |
| maximal normal subgroup of | #151514, #151515, #151516, … |
| maximal quotient of | #151714, #151718, #151719, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6715);