Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3057 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25 |
| socle | C22 | C23×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C4×#41 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
989 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)(11,17)(12,18)(14,20)(16,21)(19,23)(22,24);(2,5)(6,8)(9,10)(11,14)(12,16)(13,15)(17,20)(18,21)(19,22)(23,24);(9,11,18,23,13,19,12,17)(10,14,21,24,15,22,16,20);(1,3,4,7)(2,6,5,8);(10,15)(14,22)(16,21)(20,24)Polycyclic presentation:
2^8|c^2=gh,d^2=f,g^2=h,b^a=bf,c^a=cg,g^a=gh,e^b=eh64 conjugacy classes — view the full character table.
| maximal subgroup of | #129564, #129565, #220234, … |
| maximal normal subgroup of | #129564, #129565, #220234, … |
| maximal quotient of | #6371184, #6372115, #6372117, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 53683);