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 | |
806 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)(19,24)(20,32)(21,26)(22,36)(23,27)(25,34)(28,41)(29,38)(30,39)(31,45)(33,40)(35,46)(37,44)(42,47)(43,48);(1,3)(2,6)(4,11)(5,10)(7,14)(8,9)(12,15)(13,16)(17,19)(18,24)(20,28)(21,29)(22,31)(23,30)(25,37)(26,38)(27,39)(32,41)(33,42)(34,44)(35,43)(36,45)(40,47)(46,48);(17,20,21,33,23,35,22,34)(18,25,36,46,27,40,26,32)(19,28,29,42,30,43,31,44)(24,37,45,48,39,47,38,41);(3,10)(6,9)(11,16)(14,15)(17,21,23,22)(18,26,27,36)(19,29,30,31)(20,33,35,34)(24,38,39,45)(25,32,40,46)(28,42,43,44)(37,41,47,48);(1,4,5,13)(2,7,8,12)(3,11,10,16)(6,14,9,15)(19,30)(24,39)(28,43)(29,31)(37,47)(38,45)(41,48)(42,44)Polycyclic presentation:
2^8|a^2=f,c^2=gh,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 | #7624718, #7624719, #7624720, … |
| maximal normal subgroup of | #7624718, #7624719, #7624720, … |
| maximal quotient of | #6386887, #6389507, #6389540, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 55478);