Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3585 |
| automorphism | Aut(G) order 24576 |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C23×D4 |
| commutator | C4 | C24×C4 |
| Frattini, cosocle | C2×C4 | C25 |
| prime radical | C2×#577 | C2 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C4×#41 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
1367 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 20 points:
(1,2,4,3)(5,6)(7,12)(8,10)(9,11)(13,16)(14,18)(15,17)(19,20);(6,10)(8,16)(11,13)(12,17)(15,20)(18,19);(1,3,4,2)(6,11)(10,13)(12,18)(17,19);(5,7)(6,12)(8,15)(9,14)(10,17)(11,18)(13,19)(16,20);(7,14)(12,18)(15,20)(17,19)Polycyclic presentation:
2^8|a^2=g,c^2=g,f^2=h,b^a=bf,c^a=ch,f^a=fh,f^b=fh,e^d=eh82 conjugacy classes — view the full character table.
| Fitting subgroup of | #60174⋊C3 |
| maximal subgroup of | #6383871, #6383872, #6383873, … |
| maximal normal subgroup of | #6383871, #6383872, #6383873, … |
| maximal quotient of | #6344531, #6344571, #6344574, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 53162);