Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3585 |
| automorphism | ? |
| 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×#575 | C2 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
1371 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)(8,13)(9,18)(10,14)(11,15)(16,19)(17,20);(5,7)(6,11)(8,15)(9,17)(10,16)(12,18)(13,19)(14,20);(1,3,4,2)(5,7)(6,12)(8,17)(9,15)(10,16)(11,18)(13,20)(14,19);(5,8,10,9)(6,13,14,18)(7,15,16,17)(11,19,20,12);(6,14)(11,20)(12,19)(13,18)Polycyclic presentation:
2^8|a^2=g,c^2=g,d^2=h,f^2=h,b^a=bf,e^a=eh,f^a=fh,f^b=fh,d^c=dh82 conjugacy classes — view the full character table.
| maximal subgroup of | #364597, #364598, #364635, … |
| maximal normal subgroup of | #364597, #364598, #364635, … |
| maximal quotient of | #6344527, #6344563, #6344564, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 53158);