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 ◁ #533 ◁ 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)(19,27)(20,31)(21,25)(22,36)(23,29)(24,34)(26,35)(28,42)(30,38)(32,46)(33,45)(37,47)(39,43)(40,48)(41,44);(1,3,5,11)(2,6,8,9)(4,12,13,10)(7,15,16,14)(17,19,21,29)(18,23,25,27)(20,32,33,28)(22,30,35,44)(24,39,40,37)(26,38,36,41)(31,42,45,46)(34,47,48,43);(17,20,22,34,21,33,35,48)(18,24,26,31,25,40,36,45)(19,28,30,43,29,32,44,47)(23,37,38,46,27,39,41,42);(1,4,5,13)(2,7,8,16)(3,10,11,12)(6,14,9,15)(19,29)(23,27)(28,32)(30,44)(37,39)(38,41)(42,46)(43,47);(2,8)(6,9)(7,16)(14,15)(18,25)(19,29)(24,40)(26,36)(28,32)(30,44)(31,45)(43,47)Polycyclic presentation:
2^8|b^2=f,c^2=g,d^2=fh,g^2=h,b^a=bf,c^a=cg,e^a=ef,g^a=gh,c^b=ch,d^b=df,e^b=eh52 conjugacy classes — view the full character table.
| maximal subgroup of | #131772, #131775, #489667, … |
| maximal normal subgroup of | #131772, #131775, #489667, … |
| maximal quotient of | #6386833, #6389330, #6389332, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 55371);