Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #13694 |
| automorphism | Aut(G) order 16384 |
| 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 | C2×D32 | 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 | |
119 conjugacy classes of subgroups.
| minimal degree | μ(G) = 68 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 68 points:
(2,4)(5,6)(7,17)(8,23)(9,13)(10,29)(11,30)(12,19)(14,18)(15,24)(16,28)(20,37)(21,50)(22,51)(25,52)(26,56)(27,57)(31,46)(32,60)(33,61)(34,42)(35,55)(36,62)(38,41)(39,45)(40,49)(43,54)(44,53)(47,58)(48,67)(59,68)(63,65)(64,66);(5,7,18,41,10,21,43,61,28,49,62,68,55,66,25,46,9,20,42,65,27,48,26,47,11,22,44,60,24,45,8,19)(6,12,23,39,15,32,53,51,30,58,56,67,57,63,34,37,13,31,52,64,35,59,36,40,16,33,54,50,29,38,14,17);(1,2,3,4)(6,13)(12,31)(14,34)(15,35)(16,30)(17,37)(23,52)(29,57)(32,59)(33,58)(36,53)(38,63)(39,64)(40,51)(50,67)(54,56)Polycyclic presentation:
2^8|b^2=df,c^2=eh,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 | #94736, #94737, #153653, … |
| maximal normal subgroup of | #94736, #94737, #153653, … |
| maximal quotient of | #151712, #151713, #151714, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6716);