Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #13661 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×D16 |
| commutator, Frattini, cosocle | C2×C16 | C23 |
| 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 | |
116 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)(5,6)(7,11)(8,20)(9,23)(10,15)(12,16)(13,17)(14,21)(18,30)(19,26)(22,36)(24,32)(25,31)(27,35)(28,33)(29,34);(1,2,3,4);(6,11)(7,16)(8,17)(9,21)(12,13)(14,30)(15,26)(18,35)(19,33)(20,24)(22,34)(23,25)(27,36)(28,29)(31,32)Polycyclic presentation:
2^8|b^2=d,e^2=fg,f^2=gh,g^2=h,b^a=bd,c^a=ce,e^a=ef,f^a=fg,g^a=gh,e^c=ef,f^c=fg,g^c=gh49 conjugacy classes — view the full character table.
| maximal subgroup of | #151512, #151515, #151520, … |
| maximal normal subgroup of | #151512, #151515, #151520, … |
| maximal quotient of | #151618, #151628, #151652, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6673);