Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60367 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
2405 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)(7,15)(8,12)(9,13)(10,14)(11,16)(17,19)(18,20);(5,7)(6,11)(8,16)(9,18)(10,17)(12,19)(13,15)(14,20);(2,4)(5,6)(7,11)(8,13)(9,12)(10,14)(15,16)(17,20)(18,19);(5,8,10,9)(6,12,14,13)(7,16,17,18)(11,19,20,15);(5,8,10,9)(6,13,14,12)(7,16,17,18)(11,15,20,19)Polycyclic presentation:
2^8|d^2=h,e^2=h,f^2=h,b^a=bf,c^a=cg,e^a=eh,f^a=fh,f^b=fh,d^c=dh58 conjugacy classes — view the full character table.
| maximal subgroup of | #246320, #246322, #246323, … |
| maximal normal subgroup of | #246320, #246322, #246323, … |
| maximal quotient of | #6346655, #6346664, #6346673, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 53359);