Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #13697 |
| automorphism | Aut(G) = #72287717 (order 8192) · Inn(G) = C2×D16 · Out(G) = C2×C8×D4 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×D16 |
| commutator, Frattini, cosocle | C2×C16 | C23 |
| prime radical | C23 | D16 |
| LCS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ C22×C16 ◁ G | |
| LECS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
95 conjugacy classes of subgroups.
| minimal degree | μ(G) = 68 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 68 points:
(1,2)(3,4)(5,6,11,16)(7,17,19,40)(8,23,26,55)(9,27,29,57)(10,30,28,15)(12,31,32,20)(13,35,37,18)(14,38,39,24)(21,51,47,68)(22,52,48,43)(25,56,45,54)(33,63,62,67)(34,50,58,61)(36,46,53,64)(41,60,65,44)(42,59,66,49);(5,7,18,44,9,21,45,61,28,48,64,63,38,59,26,31,11,19,35,60,29,47,25,50,10,22,46,67,24,49,8,20)(6,12,23,42,14,33,53,52,30,58,56,68,57,65,37,40,16,32,55,66,39,62,36,43,15,34,54,51,27,41,13,17);(2,4)(7,19)(12,32)(17,40)(20,31)(21,47)(22,48)(33,62)(34,58)(41,65)(42,66)(43,52)(44,60)(49,59)(50,61)(51,68)(63,67)Polycyclic presentation:
2^8|a^2=h,b^2=df,d^2=fg,f^2=gh,g^2=h,b^a=bd,c^a=ce,d^a=df,f^a=fg,g^a=gh,c^b=ch46 conjugacy classes — view the full character table.
| maximal subgroup of | #94695, #94697, #153608, … |
| maximal normal subgroup of | #94695, #94697, #153608, … |
| maximal quotient of | #151641, #151650, #151664, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6686);