Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60470 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C22×#121 |
| commutator, Frattini, cosocle | C2×C4 | C25 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×#41 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
1315 conjugacy classes of subgroups; only the 517 normal subgroups are shown.
This group has 1315 conjugacy classes of subgroups — too many to draw. Only its 517 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) = 32 |
A minimal faithful permutation representation — 5 generators on 32 points:
(2,7)(3,12)(5,8)(10,19)(11,20)(13,26)(14,15)(17,21)(18,23)(25,30)(27,28)(29,32);(1,2,8,20,6,11,5,7)(3,9,12,23,15,24,14,18)(4,10,21,30,16,25,17,19)(13,22,26,32,27,31,28,29);(1,3)(2,9)(4,13)(5,14)(6,15)(7,18)(8,12)(10,22)(11,24)(16,27)(17,28)(19,29)(20,23)(21,26)(25,31)(30,32);(1,4)(2,10)(3,13)(5,17)(6,16)(7,19)(8,21)(9,22)(11,25)(12,26)(14,28)(15,27)(18,29)(20,30)(23,32)(24,31);(4,16)(10,25)(13,27)(17,21)(19,30)(22,31)(26,28)(29,32)Polycyclic presentation:
2^8|b^2=fh,f^2=h,g^2=h,b^a=bf,c^a=cg,f^a=fh,g^a=gh,g^c=gh,e^d=eh58 conjugacy classes — view the full character table.
| Fitting subgroup of | #60471⋊C3 |
| maximal subgroup of | #499905, #499907, #501080, … |
| maximal normal subgroup of | #499905, #499907, #501080, … |
| maximal quotient of | #6353804, #6353808, #6353809, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 53459);