Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #62307 |
| 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 ◁ #532 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
913 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,7)(4,6)(5,8)(9,10,15,19)(11,16,13,17)(12,21,24,20)(14,23,22,18);(1,3)(2,5)(4,8)(6,7)(9,11,15,13)(10,16,19,17)(12,18,24,23)(14,20,22,21);(9,12,22,17,15,24,14,16)(10,13,23,20,19,11,18,21);(3,8)(5,7)(9,13,15,11)(10,17,19,16)(12,23,24,18)(14,21,22,20);(2,6)(5,7)(11,13)(12,24)(16,17)(20,21)Polycyclic presentation:
2^8|a^2=h,b^2=h,c^2=gh,d^2=h,g^2=h,b^a=bf,c^a=cg,e^a=ef,g^a=gh,d^b=df,e^b=eh,e^c=eh,e^d=eh49 conjugacy classes — view the full character table.
| maximal subgroup of | #6367570, #6367571, #6367578, … |
| maximal normal subgroup of | #6367570, #6367571, #6367578, … |
| maximal quotient of | #6386789, #6389223, #6389231, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 55344);