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 | #2408 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1306 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 20 points:
(2,3)(6,10)(7,8)(13,14,17,19)(15,16,20,18);(1,2,4,3)(5,6,9,10)(7,11,8,12)(15,20)(16,18);(5,6)(7,11)(8,12)(9,10)(13,15)(14,18)(16,19)(17,20);(5,7,9,8)(6,11,10,12)(13,16,17,18)(14,15,19,20);(1,3,4,2)(5,8,9,7)(6,12,10,11)Polycyclic presentation:
2^8|a^2=h,b^2=f,d^2=g,e^2=f,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 | #7037187, #7037191, #7037251, … |
| maximal normal subgroup of | #7037187, #7037191, #7037251, … |
| maximal quotient of | #531635, #531680, #531774, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 52261);