Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #58990 |
| 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 |
| prime radical | #2669 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1210 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 20 points:
(5,6)(7,11)(8,12)(9,10)(14,18)(15,20);(2,4)(6,10)(7,12)(14,18)(15,20);(5,7)(6,8)(9,12)(10,11)(13,14,17,18)(15,19,20,16);(1,2,3,4)(5,7)(6,8)(9,12)(10,11)(13,15)(14,19)(16,18)(17,20);(5,8)(6,7)(9,11)(10,12)(13,16,17,19)(14,15,18,20)Polycyclic presentation:
2^8|c^2=fg,d^2=gh,e^2=fg,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=dh64 conjugacy classes — view the full character table.
| maximal subgroup of | #133571, #500761, #503186, … |
| maximal normal subgroup of | #133571, #500761, #503186, … |
| maximal quotient of | #531849, #532318, #532504, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 52834);