Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #45438 |
| automorphism | Aut(G) order 196608 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
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 | |
1455 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 20 points:
(2,4)(5,6)(7,11)(8,9)(10,12)(14,15)(18,19);(6,9)(10,11)(14,15)(16,20);(1,2,3,4)(13,14,17,15)(16,18,20,19);(5,6)(7,10)(8,9)(11,12)(13,15,17,14)(16,19,20,18);(5,7,8,12)(6,10,9,11)(13,16)(14,18)(15,19)(17,20)Polycyclic presentation:
2^8|c^2=g,d^2=h,e^2=f,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.
| Fitting subgroup of | #45992⋊C3 |
| maximal subgroup of | #7221041, #7221045, #7221105, … |
| maximal normal subgroup of | #7221041, #7221045, #7221105, … |
| maximal quotient of | #557077, #557088, #557386, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 38980);