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 | #3212 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ #533 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
827 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)(3,11)(4,8)(5,9)(6,10)(7,13)(12,16)(14,15)(17,18,22,26)(19,23,28,37)(20,30,32,40)(21,33,35,25)(24,34,39,31)(27,41,42,48)(29,45,44,38)(36,46,47,43);(1,3,6,13)(2,7,10,11)(4,12,5,14)(8,15,9,16)(17,19)(18,23)(20,27)(21,29)(22,28)(24,36)(25,38)(26,37)(30,41)(31,43)(32,42)(33,45)(34,46)(35,44)(39,47)(40,48);(1,4,6,5)(2,8,10,9)(3,12,13,14)(7,15,11,16)(17,20,22,32)(18,24,26,39)(19,27,28,42)(21,34,35,31)(23,36,37,47)(25,40,33,30)(29,46,44,43)(38,48,45,41);(1,5,6,4)(2,9,10,8)(3,12,13,14)(7,15,11,16)(19,28)(23,37)(27,42)(29,44)(36,47)(38,45)(41,48)(43,46);(2,10)(7,11)(8,9)(15,16)(19,28)(23,37)(27,42)(29,44)(36,47)(38,45)(41,48)(43,46)Polycyclic presentation:
2^8|a^2=h,b^2=f,c^2=fh,d^2=f,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 | #6364387, #6364388, #6364391, … |
| maximal normal subgroup of | #6364387, #6364388, #6364391, … |
| maximal quotient of | #6386333, #6388242, #6388245, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 54917);