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 |
| prime radical | #3521 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ #529 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
826 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,5,8)(3,9,10,6)(4,12,13,7)(11,14,16,15)(17,18)(19,29)(20,33)(21,28)(22,27)(23,26)(24,32)(25,34)(30,44)(31,39)(35,42)(36,41)(37,40)(38,45)(43,47)(46,48);(1,3)(2,6)(4,11)(5,10)(7,14)(8,9)(12,15)(13,16)(17,19)(18,24)(20,30)(21,36)(22,32)(23,31)(25,38)(26,41)(27,29)(28,39)(33,48)(34,47)(35,46)(37,45)(40,43)(42,44);(17,20)(18,25)(19,30)(21,34)(22,35)(23,37)(24,38)(26,33)(27,40)(28,42)(29,43)(31,45)(32,46)(36,47)(39,44)(41,48);(3,10)(6,9)(11,16)(14,15)(17,21,22,23)(18,26,27,28)(19,31,32,36)(20,34,35,37)(24,39,29,41)(25,33,40,42)(30,45,46,47)(38,44,43,48);(1,4,5,13)(2,7,8,12)(3,11,10,16)(6,14,9,15)(18,27)(19,32)(25,40)(26,28)(30,46)(31,36)(33,42)(45,47)Polycyclic presentation:
2^8|a^2=fh,d^2=h,e^2=fh,g^2=h,b^a=bf,c^a=cg,d^a=dh,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×#62196, #10579450, #10579521, … |
| maximal normal subgroup of | C2×#62196, #10579450, #10579521, … |
| maximal quotient of | #6386448, #6388563, #6388633, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 55184);