Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3264 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25 |
| socle | C22 | C23×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
1395 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)(7,14)(8,12)(9,18)(10,13)(11,17)(15,16)(19,20);(5,7)(6,11)(8,17)(9,19)(10,15)(12,16)(13,20)(14,18);(2,4)(7,15)(11,20)(14,16)(17,19);(5,6)(7,16)(8,12)(9,18)(10,13)(11,19)(14,15)(17,20);(5,8,10,9)(6,12,13,18)(7,17,15,19)(11,16,20,14)Polycyclic presentation:
2^8|e^2=h,f^2=h,b^a=bf,c^a=cg,f^a=fh,c^b=ch,d^b=df,f^b=fh,f^d=fh58 conjugacy classes — view the full character table.
| maximal subgroup of | #131052, #131053, #131854, … |
| maximal normal subgroup of | #131052, #131053, #131854, … |
| maximal quotient of | #6358256, #6358265, #6358278, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 54026);