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 ◁ #525 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
929 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)(11,14)(12,20)(13,17)(15,19)(16,23)(18,21)(22,24);(1,3)(2,5)(4,8)(6,7)(9,11,14,10)(12,18,22,16)(13,17,15,19)(20,23,24,21);(9,12,13,21,14,22,15,23)(10,16,19,24,11,18,17,20);(3,8)(5,7)(9,13,14,15)(10,17,11,19)(12,21,22,23)(16,20,18,24);(2,6)(5,7)(10,11)(12,22)(17,19)(21,23)Polycyclic presentation:
2^8|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=eh49 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#62320, #10577613, #10577658, … |
| maximal normal subgroup of | C2×#62320, #10577613, #10577658, … |
| maximal quotient of | #6386798, #6389161, #6389217, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 55308);