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 |
| prime radical | #3179 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ #533 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
807 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2)(3,9)(4,7)(5,8)(6,11)(10,15)(12,14)(13,16)(17,18,21,25)(19,27,29,23)(20,31,33,47)(22,35,32,26)(24,38,39,34)(28,42,44,46)(30,37,43,41)(36,45,40,48);(1,3,5,11)(2,6,8,9)(4,12,13,10)(7,15,16,14)(17,19,21,29)(18,23,25,27)(20,28,33,44)(22,30,32,43)(24,36,39,40)(26,37,35,41)(31,46,47,42)(34,45,38,48);(17,20,32,38,21,33,22,34)(18,24,35,47,25,39,26,31)(19,28,43,48,29,44,30,45)(23,36,41,42,27,40,37,46);(1,4,5,13)(2,7,8,16)(3,10,11,12)(6,14,9,15)(19,29)(23,27)(28,44)(30,43)(36,40)(37,41)(42,46)(45,48);(2,8)(6,9)(7,16)(14,15)(18,25)(19,29)(20,33)(26,35)(30,43)(34,38)(36,40)(42,46)Polycyclic presentation:
2^8|a^2=h,b^2=f,c^2=gh,d^2=fh,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×#62412, #10578448, #10578457, … |
| maximal normal subgroup of | C2×#62412, #10578448, #10578457, … |
| maximal quotient of | #6386850, #6389348, #6389406, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 55400);