Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #560 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C8 | C24 |
| commutator, socle | C22 | C24×C4 |
| Frattini, cosocle | C2×C4 | C25 |
| prime radical | #3531 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
814 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)(3,9)(4,7)(5,8)(6,11)(10,15)(12,14)(13,16)(17,18,21,26,22,24,32,37)(19,23,29,35,27,33,40,44)(20,25,30,38,31,36,42,46)(28,34,41,45,39,43,47,48);(1,3,5,11)(2,6,8,9)(4,12,13,10)(7,15,16,14)(17,19)(18,23)(20,28)(21,29)(22,27)(24,33)(25,34)(26,35)(30,41)(31,39)(32,40)(36,43)(37,44)(38,45)(42,47)(46,48);(1,4,5,13)(2,7,8,16)(3,10,11,12)(6,14,9,15)(18,24)(19,27)(25,36)(26,37)(28,39)(29,40)(38,46)(41,47);(2,8)(6,9)(7,16)(14,15)(19,27)(23,33)(28,39)(29,40)(34,43)(35,44)(41,47)(45,48);(17,20,21,30,22,31,32,42)(18,25,26,38,24,36,37,46)(19,28,29,41,27,39,40,47)(23,34,35,45,33,43,44,48)Polycyclic presentation:
2^8|a^2=g,b^2=f,c^2=f,e^2=g,g^2=h,b^a=bf,c^a=ch,d^a=df,c^b=cfh,d^b=dh76 conjugacy classes — view the full character table.
| Fitting subgroup of | #60332⋊C5 |
| maximal subgroup of | #129672, #210390, #210391, … |
| maximal normal subgroup of | #129672, #210390, #210391, … |
| maximal quotient of | #6370565, #6370654, #6377920, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 53320);