Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60388 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C22×#120 |
| commutator, Frattini, cosocle | C2×C4 | C25 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×#41 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
2385 conjugacy classes of subgroups; only the 557 normal subgroups are shown.
This group has 2385 conjugacy classes of subgroups — too many to draw. Only its 557 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:
(1,2,6,10)(3,11,14,24)(4,8,15,22)(5,17,19,9)(7,18,21,13)(12,25,26,32)(16,29,28,23)(20,30,31,27);(1,3)(2,7)(4,12)(5,18)(6,14)(8,20)(9,24)(10,21)(11,17)(13,19)(15,26)(16,30)(22,31)(23,32)(25,29)(27,28);(5,19)(9,17)(11,24)(13,18)(16,28)(23,29)(25,32)(27,30);(1,4)(2,8)(3,12)(5,16)(6,15)(7,20)(9,23)(10,22)(11,25)(13,27)(14,26)(17,29)(18,30)(19,28)(21,31)(24,32);(4,15)(8,22)(12,26)(16,28)(20,31)(23,29)(25,32)(27,30)Polycyclic presentation:
2^8|a^2=h,f^2=h,b^a=bf,c^a=cg,f^a=fh,f^b=fh,g^b=gh,f^c=fh,e^d=eh58 conjugacy classes — view the full character table.
| Fitting subgroup of | #60389⋊C3 |
| maximal subgroup of | #152025, #499882, #499883, … |
| maximal normal subgroup of | #152025, #499882, #499883, … |
| maximal quotient of | #6350413, #6350414, #6350415, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 53377);