Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61609 |
| 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 ◁ #529 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
948 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,5)(3,7)(11,19)(12,13)(15,22)(16,18)(20,21)(23,24);(1,2)(3,6)(4,5)(7,8)(9,10,14,17)(11,15,20,23)(12,16,13,18)(19,22,21,24);(9,11)(10,15)(12,19)(13,21)(14,20)(16,22)(17,23)(18,24);(2,5)(6,8)(9,12,14,13)(10,16,17,18)(11,19,20,21)(15,22,23,24);(1,3,4,7)(2,6,5,8)(10,17)(15,23)(16,18)(22,24)Polycyclic presentation:
2^8|b^2=h,d^2=h,e^2=f,g^2=h,b^a=bf,c^a=cg,d^a=dh,e^a=ef,g^a=gh,d^b=df,e^b=eh,g^c=gh49 conjugacy classes — view the full character table.
| maximal subgroup of | #6363895, #6363899, #6363927, … |
| maximal normal subgroup of | #6363895, #6363899, #6363927, … |
| maximal quotient of | #6386441, #6388562, #6388619, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 55021);