Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #534 |
| automorphism | Aut(G) order 524288 |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C24 |
| commutator | C22 | C24×C4 |
| Frattini, socle, cosocle | C23 | C25 |
| prime radical | C23×D4 | C22 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1473 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 20 points:
(1,2,3,4)(5,6)(7,11)(8,10)(9,12)(13,14)(15,18)(16,17)(19,20);(5,7)(6,9)(8,12)(10,11);(7,12)(9,11)(14,17)(18,20);(6,10)(9,11);(13,15,16,19)(14,18,17,20)Polycyclic presentation:
2^8|a^2=h,e^2=g,b^a=bf,c^a=cg,d^a=df,c^b=cf88 conjugacy classes — view the full character table.
| maximal subgroup of | #106319, #106373, #106530, … |
| maximal normal subgroup of | #106319, #106373, #106530, … |
| maximal quotient of | #513388, #513389, #513401, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 27045);