Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3055 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25 |
| socle | C22 | C23×D4 |
| prime radical | C2×#532 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
1085 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,6)(4,5)(7,8)(9,10)(11,17)(12,16)(13,18)(14,19)(15,20)(21,24)(22,23);(2,5)(6,8)(9,11,14,22,15,21,13,12)(10,16,18,24,20,23,19,17);(1,3,4,7)(2,6,5,8)(9,12,13,21,15,22,14,11)(10,17,19,23,20,24,18,16);(11,21)(12,22)(16,23)(17,24);(9,13,15,14)(10,18,20,19)(11,12,21,22)(16,24,23,17)Polycyclic presentation:
2^8|b^2=g,c^2=f,e^2=h,f^2=h,g^2=h,b^a=bf,c^a=cg,f^a=fh,g^a=gh,d^b=dh,d^c=dh64 conjugacy classes — view the full character table.
| maximal subgroup of | #111361, #111364, #111438, … |
| maximal normal subgroup of | #111361, #111364, #111438, … |
| maximal quotient of | #6371046, #6371692, #6371946, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 53606);