Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24292 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | D42 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | #3045 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
433 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2,3,4)(5,6,9,12)(7,14,18,20)(8,13,16,11)(10,19,15,17);(5,6,8,11,9,12,16,13)(7,10,17,14,18,15,19,20);(2,4)(5,6)(7,15)(8,11)(9,12)(10,18)(13,16)(14,19)(17,20);(5,7,16,19,9,18,8,17)(6,10,13,20,12,15,11,14)Polycyclic presentation:
2^8|a^2=f,b^2=e,d^2=eh,e^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,g^b=gh,d^c=dh,g^c=gh40 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#24801, #6442951, #6442953, … |
| maximal normal subgroup of | C2×#24801, #6442951, #6442953, … |
| maximal quotient of | #277359, #277360, #277365, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 17789);