Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #62283 |
| 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 | |
918 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)(3,7)(4,6)(5,8)(9,10)(11,12)(13,17)(14,18)(15,24)(16,19)(20,22)(21,23);(1,3)(2,5)(4,8)(6,7)(9,11)(10,12)(13,14)(15,20)(16,21)(17,18)(19,23)(22,24);(9,12,15,17,16,23,14,22)(10,13,18,11,19,20,24,21);(3,8)(5,7)(9,13,16,20)(10,17,19,22)(11,14,21,15)(12,18,23,24);(2,6)(5,7)(9,14,16,15)(10,18,19,24)(11,20,21,13)(12,22,23,17)Polycyclic presentation:
2^8|c^2=g,d^2=h,e^2=h,g^2=h,b^a=bf,c^a=cg,e^a=ef,g^a=gh,c^b=ch,d^b=df,e^b=eh,e^d=eh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#62338, #10577631, #10577669, … |
| maximal normal subgroup of | C2×#62338, #10577631, #10577669, … |
| maximal quotient of | #6386767, #6389177, #6389178, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 55326);