Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24269 |
| automorphism | Aut(G) = C211×D4 (order 16384) · Inn(G) = C22×D4 · Out(G) = C29 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C2×#121 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
419 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2,4,3)(5,6)(7,9)(8,10)(11,12)(13,14)(15,19)(16,20)(17,18);(1,3,4,2)(5,7,8,11)(6,9,10,12)(13,15,19,14,17,20,16,18);(2,3)(6,10)(9,12)(14,18)(15,20);(13,15,19,14,17,20,16,18)Polycyclic presentation:
2^8|a^2=gh,b^2=ef,d^2=eh,e^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,d^c=dh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#27796, #6457578, #6457592, … |
| maximal normal subgroup of | C2×#27796, #6457578, #6457592, … |
| maximal quotient of | #315056, #315060, #315064, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 20784);