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 ◁ #529 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
842 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2)(3,10)(4,8)(5,14)(6,9)(7,12)(11,16)(13,15)(17,18)(19,23)(20,31)(21,25)(22,27)(24,33)(26,32)(28,41)(29,36)(30,37)(34,39)(35,43)(38,42)(40,45)(44,46)(47,48);(1,3)(2,7)(4,11)(5,13)(6,12)(8,15)(9,10)(14,16)(17,19)(18,23)(20,28)(21,29)(22,30)(24,35)(25,36)(26,38)(27,37)(31,41)(32,42)(33,43)(34,44)(39,46)(40,47)(45,48);(1,4,6,5)(2,8,9,14)(3,11,12,13)(7,15,10,16)(17,20,32,45,22,34,21,33)(18,24,25,39,27,40,26,31)(19,28,42,48,30,44,29,43)(23,35,36,46,37,47,38,41);(3,12)(7,10)(11,13)(15,16)(17,21,22,32)(18,25,27,26)(19,29,30,42)(20,33,34,45)(23,36,37,38)(24,39,40,31)(28,43,44,48)(35,46,47,41);(1,5,6,4)(2,8,9,14)(3,13,12,11)(7,15,10,16)(19,30)(23,37)(28,44)(29,42)(35,47)(36,38)(41,46)(43,48)Polycyclic presentation:
2^8|c^2=fgh,d^2=h,e^2=f,g^2=h,b^a=bf,c^a=cg,e^a=ef,g^a=gh,d^b=df,e^b=eh52 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#62555, #10580001, #10580055, … |
| maximal normal subgroup of | C2×#62555, #10580001, #10580055, … |
| maximal quotient of | #6386952, #6389646, #6389678, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 55543);