Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24378 |
| automorphism | Aut(G) = C211×D4 (order 16384) · Inn(G) = C22×D4 · Out(G) = C29 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | #1999 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
608 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 28 points:
(5,6)(7,11)(8,9)(10,12)(13,14)(15,21)(16,18)(17,25)(19,23)(20,24)(22,27)(26,28);(1,2)(3,4)(6,9)(10,11)(14,18)(19,26)(20,25)(21,27);(2,4)(13,15,17,23,16,22,24,28)(14,19,20,21,18,26,25,27);(5,7,8,12)(6,10,9,11)(14,18)(19,26)(20,25)(21,27)Polycyclic presentation:
2^8|c^2=f,d^2=eh,f^2=h,b^a=be,c^a=cf,d^a=de,f^a=fh,c^b=cg58 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#25601, #6449118, #6449137, … |
| maximal normal subgroup of | C2×#25601, #6449118, #6449137, … |
| maximal quotient of | #293479, #293484, #293489, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 18589);