Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3484 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C2×C4 | C25 |
| commutator, socle | C22 | C24×C4 |
| prime radical | C2×#525 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
1431 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 5 generators on 20 points:
(1,2)(3,4)(5,6,8,12,9,11,15,18)(7,10,14,17,13,16,19,20);(2,4);(5,7,8,14,9,13,15,19)(6,10,12,17,11,16,18,20);(7,13)(10,16)(14,19)(17,20);(6,11)(10,16)(12,18)(17,20)Polycyclic presentation:
2^8|a^2=g,c^2=g,g^2=h,b^a=bf,e^a=eh,d^c=dh88 conjugacy classes — view the full character table.
| maximal subgroup of | #105760, #105778, #105781, … |
| maximal normal subgroup of | #105760, #105778, #105781, … |
| maximal quotient of | #6345277, #6345278, #6345279, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 53129);