Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24274 |
| 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 | C22×D4 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
414 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 20 points:
(2,3)(5,6)(7,9)(8,10)(11,12)(13,14,18,19,17,15,16,20);(1,2,4,3)(5,7,8,11)(6,9,10,12)(13,15,16,19,17,14,18,20);(6,10)(9,12)(13,15,17,14)(16,20,18,19);(1,3,4,2)(13,16,17,18)(14,19,15,20)Polycyclic presentation:
2^8|a^2=g,b^2=efh,c^2=h,d^2=e,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,c^b=cg,d^b=dh,f^c=fh,g^c=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#27785, #6457661, #6457668, … |
| maximal normal subgroup of | C2×#27785, #6457661, #6457668, … |
| maximal quotient of | #315325, #315326, #315329, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 20773);