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 | #3499 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C4×#41 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
1021 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,6)(4,5)(7,8)(9,10,15,17)(11,18,12,14)(13,16,21,23)(19,24,20,22);(2,5)(6,8)(9,11,14,10,15,12,18,17)(13,19,22,16,21,20,24,23);(1,3,4,7)(2,6,5,8)(9,12,14,17,15,11,18,10)(13,20,22,23,21,19,24,16);(9,13)(10,16)(11,19)(12,20)(14,22)(15,21)(17,23)(18,24);(13,21)(16,23)(19,20)(22,24)Polycyclic presentation:
2^8|a^2=h,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 | #60623⋊C3 |
| maximal subgroup of | #249749, #249750, #249751, … |
| maximal normal subgroup of | #249749, #249750, #249751, … |
| maximal quotient of | #6371055, #6371698, #6371953, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 53611);