Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61609 |
| 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 ◁ #533 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
830 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)(3,11)(4,8)(5,9)(6,10)(7,13)(12,16)(14,15)(17,18)(19,27)(20,31)(21,25)(22,35)(23,29)(24,33)(26,34)(28,43)(30,38)(32,40)(36,39)(37,47)(41,45)(42,46)(44,48);(1,3,6,13)(2,7,10,11)(4,12,5,14)(8,15,9,16)(17,19,21,29)(18,23,25,27)(20,28,32,44)(22,30,34,46)(24,37,39,41)(26,38,35,42)(31,48,40,43)(33,45,36,47);(1,4,6,5)(2,8,10,9)(3,12,13,14)(7,15,11,16)(17,20,21,32)(18,24,25,39)(19,28,29,44)(22,36,34,33)(23,37,27,41)(26,40,35,31)(30,47,46,45)(38,43,42,48);(1,5,6,4)(2,9,10,8)(3,12,13,14)(7,15,11,16)(19,29)(23,27)(28,44)(30,46)(37,41)(38,42)(43,48)(45,47);(2,10)(7,11)(8,9)(15,16)(18,25)(19,29)(24,39)(26,35)(28,44)(30,46)(31,40)(45,47)Polycyclic presentation:
2^8|b^2=f,c^2=f,d^2=fh,g^2=h,b^a=bf,c^a=cg,e^a=ef,g^a=gh,d^b=df,e^b=eh,g^c=gh49 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#61918, #10578602, #10578637, … |
| maximal normal subgroup of | C2×#61918, #10578602, #10578637, … |
| maximal quotient of | #6386327, #6388218, #6388224, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 54906);