Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #62265 |
| 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 | |
938 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)(10,16)(11,19)(13,15)(17,20)(21,22)(23,24);(1,3)(2,5)(4,8)(6,7)(9,10,14,16)(11,17,22,23)(12,15,18,13)(19,24,21,20);(9,11,15,19,14,22,13,21)(10,17,18,24,16,23,12,20);(3,8)(5,7)(9,12,14,18)(10,13,16,15)(11,20,22,24)(17,21,23,19);(2,6)(5,7)(9,13,14,15)(10,18,16,12)(11,21,22,19)(17,24,23,20)Polycyclic presentation:
2^8|b^2=h,c^2=g,d^2=h,e^2=h,g^2=h,b^a=bf,c^a=cg,e^a=ef,g^a=gh,d^b=df,e^b=eh,e^d=eh52 conjugacy classes — view the full character table.
| maximal subgroup of | #130233, #130241, #263404, … |
| maximal normal subgroup of | #130233, #130241, #263404, … |
| maximal quotient of | #6386762, #6389167, #6389314, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 55319);