Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60457 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25 |
| prime radical | #3498 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C4×#41 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
1089 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)(3,6)(4,5)(7,8)(10,16)(11,18)(13,14)(17,23)(19,24)(21,22);(2,5)(6,8)(9,10,14,11,15,18,13,16)(12,17,22,19,20,24,21,23);(1,3,4,7)(2,6,5,8)(9,11,13,10,15,16,14,18)(12,19,21,17,20,23,22,24);(9,12)(10,17)(11,19)(13,21)(14,22)(15,20)(16,23)(18,24);(12,20)(17,24)(19,23)(21,22)Polycyclic presentation:
2^8|b^2=g,c^2=f,f^2=h,g^2=h,b^a=bf,c^a=cg,f^a=fh,g^a=gh,e^d=eh64 conjugacy classes — view the full character table.
| Fitting subgroup of | #60622⋊C3 |
| maximal subgroup of | #249739, #249740, #249741, … |
| maximal normal subgroup of | #249739, #249740, #249741, … |
| maximal quotient of | #6371055, #6371699, #6371954, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 53610);