Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61559 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25 |
| prime radical | D4×#41 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C4×#41 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
1015 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,4,7)(3,5,8,6)(9,10,13,16)(11,17,18,14)(12,20,21,15)(19,22,24,23);(9,11)(10,14)(12,19)(13,18)(15,22)(16,17)(20,23)(21,24);(2,5)(3,8)(6,7);(11,18)(14,17)(19,24)(22,23);(9,12,13,21)(10,15,16,20)(11,19,18,24)(14,22,17,23)Polycyclic presentation:
2^8|a^2=fh,e^2=f,g^2=h,b^a=bf,c^a=cg,e^a=ef,g^a=gh,d^b=df,g^c=gh61 conjugacy classes — view the full character table.
| Fitting subgroup of | #62096⋊C3 |
| maximal subgroup of | #250969, #250973, #250977, … |
| maximal normal subgroup of | #250969, #250973, #250977, … |
| maximal quotient of | #6386398, #6388401, #6388703, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 55084);