Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2403 |
| automorphism | Aut(G) order 131072 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C24 |
| commutator, Frattini, socle, cosocle | C23 | C25 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1307 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 20 points:
(1,2,3,4)(5,6)(7,10)(8,9)(11,12);(2,4)(13,14)(15,19)(16,18)(17,20);(6,9)(10,12)(13,15,16,20)(14,17,18,19);(14,18)(17,19);(5,7,8,11)(6,10,9,12)Polycyclic presentation:
2^8|a^2=f,c^2=h,e^2=g,b^a=bf,c^a=cg,c^b=ch,d^b=dh76 conjugacy classes — view the full character table.
| maximal subgroup of | #112892, #194982, #194995, … |
| maximal normal subgroup of | #112892, #194982, #194995, … |
| maximal quotient of | #514189, #514222, #514403, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 27864);