Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #13694 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | 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 | |
115 conjugacy classes of subgroups.
| minimal degree | μ(G) = 68 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 68 points:
(2,4)(5,6,11,16)(7,17,22,43)(8,23,26,55)(9,27,29,57)(10,30,28,15)(12,31,34,19)(13,35,37,18)(14,38,39,24)(20,49,51,68)(21,52,50,42)(25,56,45,54)(32,62,63,67)(33,48,58,61)(36,46,53,64)(40,60,65,44)(41,59,66,47);(5,7,18,44,9,20,45,61,28,50,64,62,38,59,26,31,11,22,35,60,29,51,25,48,10,21,46,67,24,47,8,19)(6,12,23,41,14,32,53,52,30,58,56,68,57,65,37,43,16,34,55,66,39,63,36,42,15,33,54,49,27,40,13,17);(1,2,3,4)Polycyclic presentation:
2^8|a^2=h,b^2=df,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 | #94735, #94739, #151518, … |
| maximal normal subgroup of | #94735, #94739, #151518, … |
| maximal quotient of | #151713, #151719, #151725, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6714);