Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #36702 |
| automorphism | Aut(G) order 65536 |
| 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 | |
1101 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 20 points:
(2,4)(6,10)(8,12)(13,14)(15,19)(16,18)(17,20);(6,10)(7,11)(13,14,16,18)(15,17,20,19);(1,2,3,4)(5,6,9,10)(7,8,11,12)(13,14)(15,17)(16,18)(19,20);(5,7,9,11)(6,8,10,12)(13,15)(14,17)(16,20)(18,19);(5,8)(6,11)(7,10)(9,12)Polycyclic presentation:
2^8|b^2=f,c^2=g,d^2=fh,b^a=bf,c^a=cg,d^a=df,e^a=efh,c^b=ch,d^b=dfh52 conjugacy classes — view the full character table.
| maximal subgroup of | #97257, #167773, #167774, … |
| maximal normal subgroup of | #97257, #167773, #167774, … |
| maximal quotient of | #553751, #554013, #554322, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 30118);