Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61591 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C4×Q8 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
987 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,5)(3,6)(7,8)(9,10)(11,19)(12,16)(13,17)(14,18)(15,21)(20,24)(22,23);(1,2,4,5)(3,6,7,8)(9,11,14,21)(10,15,18,19)(12,20,13,22)(16,23,17,24);(1,3,4,7)(2,6,5,8)(9,12,14,13)(10,16,18,17)(11,20,21,22)(15,23,19,24);(9,13,14,12)(10,17,18,16)(11,20,21,22)(15,23,19,24);(1,2,4,5)(3,6,7,8)(10,18)(15,19)(16,17)(23,24)Polycyclic presentation:
2^8|b^2=f,c^2=f,d^2=fh,e^2=h,g^2=h,b^a=bf,c^a=cg,e^a=ef,g^a=gh,d^b=dfh,g^c=gh52 conjugacy classes — view the full character table.
| Fitting subgroup of | #61903⋊C3 |
| maximal subgroup of | #6362217, #6362219, #6362248, … |
| maximal normal subgroup of | #6362217, #6362219, #6362248, … |
| maximal quotient of | #6386324, #6388213, #6388301, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 54891);