Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61629 |
| 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 | #3518 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ #529 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
820 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,5,8)(3,9,10,6)(4,12,13,7)(11,14,16,15)(17,18)(19,28)(20,33)(21,26)(22,27)(23,36)(24,31)(25,34)(29,44)(30,43)(32,39)(35,40)(37,41)(38,47)(42,45)(46,48);(1,3)(2,6)(4,11)(5,10)(7,14)(8,9)(12,15)(13,16)(17,19)(18,24)(20,29)(21,30)(22,31)(23,32)(25,38)(26,39)(27,28)(33,48)(34,45)(35,46)(36,43)(37,47)(40,44)(41,42);(17,20,22,35)(18,25,27,41)(19,29,31,46)(21,34,23,37)(24,38,28,42)(26,40,36,33)(30,45,32,47)(39,44,43,48);(3,10)(6,9)(11,16)(14,15)(17,21,22,23)(18,26,27,36)(19,30,31,32)(20,34,35,37)(24,39,28,43)(25,40,41,33)(29,45,46,47)(38,44,42,48);(1,4,5,13)(2,7,8,12)(3,11,10,16)(6,14,9,15)(18,27)(19,31)(25,41)(26,36)(29,46)(30,32)(33,40)(45,47)Polycyclic presentation:
2^8|a^2=fh,c^2=h,d^2=h,e^2=fh,g^2=h,b^a=bf,c^a=cg,e^a=ef,g^a=gh,d^b=dfh,e^b=eh,g^c=gh52 conjugacy classes — view the full character table.
| maximal subgroup of | #6364565, #6364569, #6364597, … |
| maximal normal subgroup of | #6364565, #6364569, #6364597, … |
| maximal quotient of | #6386460, #6388583, #6388597, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 55208);