Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #7398 |
| automorphism | Aut(G) = #72285576 (order 2048) · Inn(G) = #1343 · Out(G) = C24 |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #1343 |
| commutator | C22×C8 | C2×C4 |
| Frattini, cosocle | #408 | C22 |
| prime radical | C2×#41 | D4 |
| LCS | G ▷ C22×C8 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C22×C4 ◁ #408 ◁ G | |
| LECS | G ▷ #408 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
117 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 2 generators on 64 points:
(1,2,5,10,7,11,25,40)(3,13,15,43,17,44,48,58)(4,18,21,38,23,9,32,35)(6,12,26,39,20,34,53,57)(8,28,29,19,30,51,59,52)(14,31,46,37,47,41,60,63)(16,33,49,61,27,50,24,55)(22,42,54,64,45,36,56,62);(1,3,7,17)(2,8,11,30)(4,19,23,52)(5,24,25,49)(6,27,20,16)(9,33,18,50)(10,37,40,63)(12,41,34,31)(13,42,44,36)(14,45,47,22)(15,26,48,53)(21,46,32,60)(28,56,51,54)(29,57,59,39)(35,58,38,43)(55,62,61,64)Polycyclic presentation:
2^8|a^2=d,b^2=h,c^2=fh,d^2=h,f^2=h,b^a=bc,c^a=ce,e^a=eg,g^a=gh,c^b=cf,d^b=def,f^b=fh,d^c=dg,e^d=eh25 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#7401, #150179, #150219, … |
| maximal normal subgroup of | C2×#7401, #150179, #150219, … |
| maximal quotient of | #93208, #93212, #93224, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 389);