Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #62243 |
| 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 | C2×#41 | D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C4×#41 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
871 conjugacy classes of subgroups; only the 487 normal subgroups are shown.
This group has 871 conjugacy classes of subgroups — too many to draw. Only its 487 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,8)(3,9,10,6)(4,12,13,7)(11,14,16,15)(17,18,21,24)(19,25,28,32)(20,29,26,23)(22,30,31,27);(1,3)(2,6)(4,11)(5,10)(7,14)(8,9)(12,15)(13,16);(17,19,26,30,21,28,20,27)(18,22,29,32,24,31,23,25);(3,10)(6,9)(11,16)(14,15);(1,4,5,13)(2,7,8,12)(3,11,10,16)(6,14,9,15)Polycyclic presentation:
2^8|a^2=fh,c^2=gh,e^2=f,g^2=h,b^a=bf,c^a=cg,e^a=ef,g^a=gh,d^b=df64 conjugacy classes — view the full character table.
| Fitting subgroup of | #62472⋊C3 |
| maximal subgroup of | #131684, #131689, #250192, … |
| maximal normal subgroup of | #131684, #131689, #250192, … |
| maximal quotient of | #6386874, #6389485, #6389749, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 55460);