Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #45438 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C25 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
910 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,4)(5,6)(7,8)(10,11)(14,15)(19,23)(21,24);(2,6)(3,7)(10,11)(12,16)(17,18,20,22)(19,21,23,24);(9,10,13,11)(12,14,16,15)(17,19,20,23)(18,21,22,24);(1,3,5,7)(2,4,6,8)(9,11,13,10)(12,15,16,14);(1,4)(2,7)(3,6)(5,8)(9,12,13,16)(10,14,11,15)Polycyclic presentation:
2^8|b^2=gh,c^2=g,d^2=fh,e^2=h,b^a=bf,c^a=cg,d^a=dh,e^a=ef,c^b=ch,d^b=dfh,e^b=eh52 conjugacy classes — view the full character table.
| maximal subgroup of | #97212, #172277, #172278, … |
| maximal normal subgroup of | #97212, #172277, #172278, … |
| maximal quotient of | #557085, #557117, #557370, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 39302);