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) = C27×C4 |
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 | |
1247 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)(7,11)(10,12)(13,14)(15,19)(16,18)(17,20);(6,9)(10,12)(13,14,16,18)(15,17,20,19);(1,2,3,4)(5,6)(7,10)(8,9)(11,12)(13,14)(15,17)(16,18)(19,20);(5,6)(7,10)(8,9)(11,12)(13,15,16,20)(14,17,18,19);(5,7)(6,10)(8,11)(9,12)Polycyclic presentation:
2^8|b^2=f,c^2=g,d^2=f,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 | #98058, #261203, #261207, … |
| maximal normal subgroup of | #98058, #261203, #261207, … |
| maximal quotient of | #553758, #553767, #554020, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 29729);